Transition Tori in the Planar Restricted Elliptic
Three Body Problem
Maciej J. Capiński
Faculty of Applied Mathematics, AGH University of Science and Technology,
al. Mickiewicza 30, 30-059 Kraków, Poland.
E-mail: [email protected]
Piotr Zgliczyński
Institute of Computer Science, Jagiellonian University,
Lojasiewicza 6, 30–348 Kraków, Poland.
E-mail: [email protected]
Abstract. We consider the elliptic three body problem as a perturbation of the
circular problem. We show that for sufficiently small eccentricities of the elliptic
problem, and for energies sufficiently close to the energy of the libration point L2 ,
a Cantor set of Lyapounov orbits survives the perturbation. The orbits are perturbed
to quasi-periodic invariant tori. We show that for a certain family of masses of
the primaries, for such tori we have transversal intersections of stable and unstable
manifolds, which lead to chaotic dynamics involving diffusion over a short range of
energy levels. Some parts of our argument are nonrigorous, but are strongly backed
by numerical computations.
8 November 2010
† The research of the first author was partially supported by the Polish Ministry of Science and Higher
Education.
† The second author was supported in part by Polish State Ministry of Science and Information
Technology grant N201 024 31/2163
† Both authors are supported by the Polish State Ministry of Science and Information Technology
grant N201 543238.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
2
1. Introduction
In the planar restricted circular three body problem (PRC3BP) two large masses µ
and 1 − µ rotate on planar circular Keplerian orbits. For convenience we will call
the larger body - the Sun and the smaller massive body - the planet The problem
deals with the motion of a third massless particle (the comet or the spacecraft), which
moves on the same plane as the two larger bodies under their gravitational pull. This
problem was considered by Llibre, Martinez and Simo in [20] for energies of solutions
close to the energy of the libration point Lµ2 . There it has been shown that there exists
a family of parameters {µk }∞
k=2 for which we have a homoclinic orbit to the libration
µk
point L2 . Moreover, it has been shown that for µ close to any of the values µk , for
a Lyapounov orbit around Lµ2 with energy sufficiently close to the energy of Lµ2 , its
stable and unstable manifolds intersect transversally. This dynamics is restricted to
a constant energy manifold and leads to a homoclinic tangle and symbolic dynamics.
Later a similar problem has been numerically investigated by Koon, Lo, Marsden and
Ross [17], where smaller energies were considered. In such a case the chaotic dynamics
is extended to include homoclinic and heteroclinic tangles along stable and unstable
manifolds of Lyapounov orbits around both the libration point Lµ1 and Lµ2 . This has
been later proven by Wilczak and Zgliczyński using a method of covering relations and
rigorous computer assisted computations in [28, 29], for the case of the Sun-Jupiter
system and the energy of the comet Oterma.
All of the above mentioned results have a common feature: since the problem follows
from an autonomous Hamiltonian, the transversality of the intersections and the chaotic
dynamics of the system are always restricted to a constant energy manifold. In this paper
we are going to consider the planar restricted elliptic three body problem (PRE3BP),
where the equations are no longer autonomous, which means that a change of energy
of solutions is possible. We will consider the circular problem considered by Llibre,
Martinez and Simo in [20] and generalize it to allow the orbits of the planet and the
Sun to be elliptic with small eccentricities e. We will treat this as a perturbation of the
circular case. We will show that most of the Lyapounov orbits around Lµ2 persist under
such perturbation as KAM-tori. Moreover, we will show that the symbolic dynamics
associated with these orbits also survives. This kind of the ’structural stability’ of
symbolic dynamics constitute the main result of this paper. It will turn out that we
also have chaotic diffusion along the energy level. In effect the dynamics of the elliptic
problem is by one dimension richer than the dynamics of the circular problem, where
all solutions are restricted to a constant energy manifold.
The diffusion between energies discussed in this paper follows from a mechanism
similar to the 1964 Arnold’s example [2]. Arnold conjectured that this phenomenon
appears in the three body problem. The result of this paper is a small step towards a
proof of this conjecture, but the described dynamics does not fulfill all requirements.
First of all, prior to perturbation we do not have a fully integrable system. We start with
the circular problem with a setting in which we already have a transversal homoclinic
Transition Tori in the Planar Restricted Elliptic Three Body Problem
3
connection between Lyapounov orbits, which in our setting play the role of lower
dimensional normally hyperbolic invariant tori. Such systems are referred to as ”a priory
unstable”, or even ”a priory chaotic”. Secondly, and most importantly, our diffusion is
¡
¢1/2
between energies with distance of order eµ1/3
and not of order one.
Throughout some of the so far explored examples a certain pattern can be observed
in the methods used for problems involving diffusion in priory unstable systems (see for
example [22] for a result of Moeckel on detection of transition tori in the case of the
planar five body problem; or the work of Delshams, Llave and Seara [8] on diffusion
of energy for perturbations of geodesic flows on a two dimensional torus; also Wiggins
[25], [26] discusses this mechanism in the case of perturbations of completely integrable
systems). First a normally hyperbolic invariant manifold foliated by invariant tori is
found. The tori are required to have hyperbolic stable and unstable manifolds and a
transversal intersection of these manifolds. Secondly a perturbation of the system is
considered. By perturbation theory ([15], [27]) of normally hyperbolic manifolds, the
normally hyperbolic invariant manifold and its stable and unstable manifolds persist
under the perturbation. Next step is to show that on the perturbed invariant manifold
most of the invariant tori survive. This under appropriate nondegeneracy conditions is
a result of the Kolmogorov Arnold Moser Theory (KAM) [3],[16]. Using KAM-technics
(for example [14], [15] or [31]) it can be shown that most of the invariant tori persist
and form a Cantor set having a positive measure in the invariant manifold. The last
step is to show that the stable and unstable manifolds of the surviving tori intersect
transversally. This can be done by the use of a Melnikov type method along a homoclinic
orbit of the unperturbed problem. The transversal intersections between the invariant
manifolds of the perturbed tori lead to homoclinic tangles for each of the surviving tori.
In addition to this we also have a chaotic diffusion along the Cantor set of homoclinic
tangles between the tori. In this paper we will follow this procedure.
When applying the method to prove the existence of transition chains for a given
physical problem the steps of the above described procedure, which present the biggest
obstacles are usually the verification of the assumptions of the KAM theorem and
computation of the Melnikov integral.
The fundamental role for our investigation is played by the Hill’s problem. In
the neighborhood of the libration point the PCR3BP and PER3BP, when written in a
suitably rescaled Hill’s coordinates, are perturbations of the Hill’s problem depending on
two small parameters µ and e. This gives us a ’local’ picture around Lµ2 , the existence of
normally hyperbolic invariant manifold and KAM-tori. In our case the twist property
needed for the KAM theorem follows from the Lyapounov-Moser Theorem [23]. For
sufficiently small µ we will prove the twist property for the family of periodic orbits
around Lµ2 , by approximating the PRC3BP with the Hill’s problem and thus obtain the
following theorem (for a detailed formulation see Theorem 31).
Theorem 1 There exist positive constants RHill , κ, µ∗ ∈ R such that for all mass
parameters µ < µ∗ and any perturbation e such that eµ−2/3 < κ most Lyapounov orbits
with radii in Hill’s coordinates not exceeding RHill are perturbed to quasi periodic orbits
Transition Tori in the Planar Restricted Elliptic Three Body Problem
4
(in other words, to invariant two dimensional invariant tori in extended phase space).
The set of radiuses for which the tori survive forms a Cantor set with complement
¢1/2
¡
measure smaller than O( eµ−1/3
).
Our second result concerns the study of the stable and unstable leaves of the Cantor
set of surviving KAM tori. We do so by applying a modification of a Melnikov method
to obtain the following result (for a detailed formulation see Theorem 46).
Theorem 2 Assume that for the sequence of masses {µk }∞
k=2 (the sequence is specified
in Theorem 3) the twist condition holds and that a derivative of a Melnikov integral
(93) at zero is nonzero, then for any given µk there exists a radius R(µk ) such that for
−2/3
−1/3
perturbations e with eµk
< κ and sufficiently small eµk
there exists a homoclinic
and a heteroclinic tangle between the surviving tori or radii smaller than R(µk ). The
tangle implies existence of symbolic dynamics and diffusion in energy. The diffusion
³
´1/2
−1/3
occurs between surviving tori on an interval of energies of order eµk
.
To apply Theorem 2 we need to back the argument by numerical verification of its
assumptions. We have verified the twist condition numerically for large masses µk and
provided a rigorous argument that for sufficiently small masses the condition holds true.
For the Melnikov integral (93) we can rigorously prove that it is convergent and that it
is zero at zero. It needs to be stressed though that the assumption that the derivative
of the Melnikov integral at zero is nonzero has only been verified numerically.
We believe that the above mentioned numerical computations can be performed
using an rigorous-computer-assisted approach in the spirit of [28]. Such arguments
require careful estimates, use of topological tools, and are the subject of ongoing work.
In our work we have been unable to obtain uniform bounds for the size of the radii
R(µk ) from Theorem 2 with respect to µk . From our proof it only follows that these
−1/3
need to decrease together with µk so that at least µk R(µk ) is smaller than some
constant. It is possible though that in a number of needed estimates R(µk ) has to be
chosen even smaller. We remark also that the symbolic dynamics proved in Theorem
2 will hold not only for the family of parameters {µk }, but also for other masses µ for
which |µk − µ| < ²k with sufficiently small ²k .
The paper is organized as follows. Section 2 contains preliminaries, where we recall
the earlier results on the planar restricted three body problem of [20], and introduce
basic facts about the Hill’s problem and the PRE3BP. In Section 3 we present the
Lyapounov–Moser theorem [23] and show how to apply it to obtain the twist property.
In Section 4 we show that we have a twist on the family of Lyapounov orbits around
Lµ2 . In Section 5 we apply the normally hyperbolic invariant manifold theorem together
with the KAM Theorem to show that most of the Lyapounov orbits around Lµ2 persist
under perturbation from the circular problem to the elliptic problem and prove the first
of our two main theorems. In Section 6 we use a Melnikov type argument to detect the
transversal intersections between the stable and unstable manifolds of the perturbed
Lyapounov orbits. In Section 7 we compute the Melnikov integral. In Section 8 we
gather together our results and prove Theorem 2.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
5
2. Preliminaries
2.1. The Planar Restricted Circular Three Body Problem
In the planar restricted circular three body problem (PRC3BP) we consider the motion
of a small massless particle (a comet or a spacecraft), under the gravitational pull of
two larger bodies of mass µ and 1 − µ (called the planet and the Sun, respectively)
which move around the origin on circular orbits of period 2π on the same plane as the
massless body. The Hamiltonian of the problem is given by [1]
H(µ, q, p, t) =
p21 + p22 1 − µ
µ
−
−
,
2
r1 (t)
r2 (t)
(1)
where (p, q) = (q1 , q2 , p1 , p2 ) are the coordinates of the massless particle and r1 (t) and
r2 (t) are the distances from the masses 1 − µ and µ respectively. After introducing a
new coordinates system (x, y, px , py )
x = q1 cos t + q2 sin t,
y = −q1 sin t + q2 cos t,
px = p1 cos t + p2 sin t,
py = −p1 sin t + p2 cos t,
(2)
which rotates together with the two larger masses, the larger masses become motionless
and one obtains [1] an autonomous Hamiltonian
H(µ, x, y, px , py ) =
(px + y)2 + (py − x)2
− Ω(x, y),
2
(3)
where
x2 + y 2 1 − µ
µ
+
+ ,
2
r1
r2
p
p
2
2
r1 = (x − µ) + y , r2 = (x + 1 − µ)2 + y 2 .
Ω(x, y) =
(4)
The motion of the particle is given by the equation
ẋ = J∇H(µ, x),
(5)
Ã
!
0
Id
where x = (x, y, px , py ) ∈ R4 , J =
and Id is a two dimensional identity
−Id 0
matrix.
The movement of the flow (5) is restricted to the hypersurfaces determined by the
energy level C,
M (µ, C) = {(x, y, px , py ) ∈ R4 |H(µ, x, y, px , py ) = C}.
(6)
In the x, y coordinates this means that the movement is restricted to the so called Hill’s
region defined by
R(µ, C) = {(x, y) ∈ R2 |Ω(x, y) ≥ −C}.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
y
y
y
a
b
c
S
S
x
6
x
L2
L1
x
z.v.c.
Figure 1. The Hill’s region for various energy levels: when the energy C is smaller
than C2µ (a), when C = C2µ (b) and when C = C1µ > C2µ (c).
The shape of the Hill’s region R(µ, C) will differ with C (see Figure 1). The focus of
our attention in this paper will be on the case when the energy C is equal to or slightly
larger than C2µ . For the energy C equal to C2µ we have the libration point Lµ2 which is
of the form (−k, 0, 0, −k) with k > 0. We shall investigate the dynamics inside of the
inner part of the Hill’s region to the right of the point Lµ2 . We shall refer to it as the
”Sun region” (see Figure 1). The boundary of this region (see Figures 1 and 2) is a
zero velocity curve (z.v.c.). The linearized vector field at the point Lµ2 has two real and
two purely imaginary eigenvalues, thus it follows [20] from the Lyapounov theorem that
for energies C larger and sufficiently close to C2µ there exists a family of periodic orbits
lµ (C) emanating from the equilibrium point Lµ2 .
L2
L2
Z.v.c.
Figure 2. The unstable manifold of Lµ2 in the x, y coordinates.
The PRC3BP admits the following reversing symmetry
S(x, y, px , py ) = (x, −y, −px , py ).
(7)
We will say that an orbit q(t) is S-symmetric when
S(q(t)) = q(−t).
(8)
In PRC3BP the Lyapounov orbits are S-symmetric (we have to choose the initial time
so that orbits start from the section {y = 0} at time t = 0).
We have the following results about the stable and unstable manifolds of Lµ2 and
lµ (C).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
7
Theorem 3 ([20, Theorem A]) For µ sufficiently small the branch of WLuµ contained
2
in the Sun region (see Figures 1 and 2) has a projection on the bounded component of
R(µ, C) given by
µ
¶
2
1/3
1/6
d(t) = µ
N (∞) − 3 + M (∞) cos t + o(1) ,
(9)
3
α(t) = −π + µ1/3 (N (∞)t + 2M (∞) sin t + o(1)) ,
(10)
where d is the distance to the z.v.c., α the angular coordinate, N (∞) and M (∞) are
constants and the expressions remain true out of a given neighborhood of Lµ2 . The
parameter t means the physical time from a suitable origin. The terms o(1) tend to
zero when µ does and they are uniform in t for t = O(µ−1/3 ).
In particular the first intersection with the x axis is orthogonal to that axis, giving
a S-symmetric homoclinic orbit for a sequence of values µ which has the following
asymptotic expression:
1
µk =
(1 + o(1)).
(11)
N (∞)3 k 3
Let us now introduce a notation for the S-symmetric homoclinic orbit to Lµ2 k
obtained in Theorem 3 for the parameters µk given in (11). We will denote such an
orbit by qµ0 k (t) (see Figure 2). We assume that such an orbit starts at a section {y = 0}
at time t = 0.
Theorem 4 ([20, Theorem B]) For µ and ∆C = C − C2µ sufficiently small, the
branch of W u (lµ (C)) contained in the Sun region intersects the plane y = 0 for x > 0
in a curve diffeomorphic to a circle (see Figure 3) given by
√
x = xw − ∆C (N + 2M cos τ )−1 (2M + N cos Mf ) (K1 cos τ cos σ − K2 sin τ sin σ)
+ µ1/3 M (1 − cos Mf )
½
¾
2M N
2
M
2/3
2
2
+µ
−
(1 − cos Mf ) + M sin Mf − N α − α cos Mf + O(µ),
3
9
3
√
−1
ẋ = sin Mf [ ∆CN (N + 2M cos τ ) (K1 cos τ cos σ − K2 sin τ sin σ)
½
¾
MN
M
1/3
2/3
2
+µ M +µ
+ 2M cos Mf + α ] + O(µ)
3
3
where xw , M, N, τ, K1 , K2 are suitable constants, α measures the distance from µ to
some µk given by Theorem 3, Mf is obtained implicitly from
µ
¶
√
1
π
−1
−2/3
Nα + µ
∆C3M (N + 2M cos τ ) (K1 cos τ cos σ − K2 sin τ sin σ)
3
N
+N Mf + 2M sin Mf = o(1),
(12)
and σ, ranging from 0 to 2π, is the parameter of the curve.
Moreover for points in the (µ, C) plane such that there exists a µk of Theorem 3
for which
4/3
∆C > Lµk (µ − µk )2
(13)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
8
holds (where L is a constant), there exist S-symmetric transversal homoclinic orbits. In
particular, for µ = µk there exist symmetrical transversal homoclinic orbits qµ0 k C for the
periodic orbit lµk (C) for C > C2µk arbitrarily close to C2µk .
Remark 5 In the original version of Theorems 3 and 4 the C was taken as the Jacobi
constant C = F where
¡
¢
F (x, y, px , py ) = −2H(µ, x, y, px , py ) = 2Ω(x, y) − ẋ2 + ẏ 2 .
(14)
In this paper we have rewritten the Theorems with C as the Hamiltonian of the PRC3BP,
which means that we have a change of sign in C compared with the original version.
Remark 6 Using a standard dynamical system theory argument, from the BirkhoffSmale homoclinic theorem, the transversal homoclinic connections to Lyapounov orbits
imply chaotic symbolic dynamics of the system. This is a content of Theorem C in
[20]. Since the system is autonomous this dynamics is restricted to the constant energy
manifold.
Remark 7 For sufficiently small µ = µk , the curves (obtained in Theorem 4) on
{y = 0} associated with the stable an unstable manifolds of lµk (C) intersect transversally
1/3
at an angle O(µk ). This can be derived based on the parameterization of the curves
from Theorem 4 combined with the symmetry property (8) of the PRC3BP. This is done
in the Appendix. For more details on the interpretation of the curves from the theorem
see also [20].
Figure 3. The intersections of the stable and unstable manifolds of lµ (C) in the
PRC3BP.
2.2. The Hill’s Problem
Let us consider a change of coordinates which shifts the origin to the smaller body of the
mass µ and rescales the coordinates by the factor µ−1/3 . We will refer to the following
as the Hill’s coordinates.
x̄ = µ−1/3 (x − (µ − 1, 0, 0, µ − 1)) .
(15)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
9
We will rewrite the Hamiltonian (3) and derive the formula of the Hill’s problem and
list a number of facts which will be relevant for us in the future.
Let us start with a simple lemma.
Lemma 8 Consider a Hamiltonian H : Rn × Rn → R and a transformation p̄ = βp,
q̄ = βq. Then (p(t), q(t)) is a solution of the Hamiltonian system for H(p,³q) if´and only
if (p̄(t), q̄(t)) is a solution of the Hamiltonian system for H̄(p̄, q̄) = β 2 H βp̄ , βq̄ .
The shift of the origin present in the transformation (15) is clearly canonical hence
we can use Lemma 8 to obtain the Hamiltonian in new variables with β = µ−1/3
¡
¢
H̄(µ, x̄) = µ−2/3 H µ, µ1/3 x̄ + (µ − 1, 0, 0, µ − 1) .
(16)
By expanding Ω in the new coordinates around zero we can rewrite our family of
Hamiltonians as
H̄(µ, x̄) = H̄(µ, x̄, ȳ, p¯x , p¯y ) =
(p̄x + ȳ)2 + (p̄y − x̄)2
+
2
(17)
1 3 2
− x̄ + O(µ1/3 ) + C(µ), for r̄ < αµ−1/3
r̄ 2
p
¡
¢
where C(µ) = µ−2/3 (1 − µ) + (1 − µ)2 /2 and r̄ =
x̄2 + ȳ 2 . The term O(µ1/3 )
depends on (x̄, ȳ) and can be written as a function a(µ, x̄, ȳ), which for any sufficiently
small α < 1 and µ ∈ [0, 1], r̄ < αµ−1/3 satisfies
−
|a(µ, x̄, ȳ)|
≤ M (α).
r3
The reason for introducing α is, that we have to be away from the Sun in order for
Taylor series of r11 to be convergent. Observe that we can drop the term C(µ).
It should be stressed that H̄ depends analytically on x̄ and µ1/3 , hence the
derivatives of the O(µ1/3 ) with respect to x̄ are still O(µ1/3 ). Therefore for fixed µ
x̄0 = J∇H̄(µ, x̄) = J∇H̄(0, x̄) + O(µ1/3 ).
The term O(µ1/3 ) is uniform in x̄ for |x̄| ≤ αµ−1/3 for any fixed sufficiently small α < 1.
The Hamiltonian H̄(0, x̄) is the Hamiltonian of the Hill’s problem
H Hill (x̄) = H̄(0, x̄).
(18)
If we denote by q Hill (t) the solution of the Hill problem and by q µ (t) the solution
of the PCR3BP, both expressed in Hill’s coordinates (15) and both starting from the
same initial condition, then the following holds
|q Hill (t) − q µ (t)| ≤ el|t| O(µ1/3 ),
(19)
provided there exist 0 < α < 1 and a compact convex set Z ⊂ B(0, αµ−1/3 ) × R2 ,
such that Z contains both q Hill ([0, t]) and q µ ([0, t]) and (0, 0) ∈
/ πx,y (Z). The constant l
depends on Z.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
10
Let us now list a few properties of the Hill’s problem. The problem has two
equilibrium points, LHill
= (−3−1/3 , 0, 0, −3−1/3 ) and LHill
= (3−1/3 , 0, 0, 3−1/3 ). The
1
2
linearization of x0 = J∇H Hill at LHill
is given by x0 = Ax, where
2


0
1
1 0
 −1 0
0 1 


A=
(20)
.
0
0 1 
 8
0 −4 −1 0
p
p
√
√
The eigenvalues of A are: ±α1 , ±α2 with α1 = 1 + 2 7 and α2 = 1 − 2 7. The
Hill problem has a reversing symmetry S given by (7).
2.3. The Planar Restricted Elliptic Three Body Problem
The planar restricted elliptic three body problem (PRE3BP) differs from the PRC3BP
by the fact that the two larger bodies move on elliptic orbits of eccentricities e instead
of circular orbits. The period of these orbits is 2π and the Hamiltonian of the PRE3BP
is analogous to (1), with the only difference that in r1 (t) and r2 (t) we take the distance
from the elliptic instead of the circular orbits of the two larger masses. The trajectories
of these orbits can be written as (see [30]) ((µ − 1)x(t), (µ − 1)y(t)) for the body µ and
(µx (t) , µy (t)) for (1 − µ), where
x(t) = (1 − e cos ψ) cos ψ + O(e2 ),
y(t) = (1 − e cos ψ) sin ψ + O(e2 ),
(21)
2
ψ(t) = t + 2e sin t + O(e ).
If one changes into the rotating coordinates (2), which is a canonical transformation
(see [21]), then the Hamiltonian (1) becomes (for a detailed derivation see the Appendix)
H e (µ, x, t) = H(µ, x) + eG(µ, x, t) + O(e2 µ−2/3 ),
(22)
where H is the Hamiltonian of the PRC3BP (3), r2 is given in (4), G is 2π periodic over
t and is given by the formula
1−µ
µ
ḡ(µ − 1, x, y, t),
3 ḡ(µ, x, y, t) +
(r1 )
(r2 )3
(23)
ḡ(α, x, y, t) = α(−2y sin t + x cos t) − α2 cos t.
(24)
G=
The term O(e2 µ−2/3 ) = a(x, y, e, µ, t) satisfies
by the following conditions
< M (δ, κ, R), on the set defined
eµ−2/3 < κ
p
r2 ≥ δµ1/3 ,
x2 + y 2 ≤ R,
µ ∈ [0, 1],
r1 > δ,
|a(x,y,e,µ,t)|
e2 µ−2/3
t ∈ R,
(25)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
11
were δ > 0 measures the closest approach to the Sun and to the planet (multiplied by
µ−1/3 ) is a number around 1/2, R is the radius of the ball containing all orbits of interest
in our problem, hence we can take R = 2 and κ is sufficiently small number (for more
details on κ see the derivation in the Appendix).
We consider points (x, y) inside of the ”Sun region” (see Figure 1) which means
that r2 > 12 kLµ2 − (µ − 1, 0, 0, µ − 1)k > δµ1/3 for some δ > 0. The Hamiltonian (22)
can be rewritten in Hill’s coordinates (15) as
H̄ e (µ, x̄, t) = H̄(µ, x̄) + eḠ (µ, x̄, t) + O(e2 r̄2 ),
(26)
Ḡ (µ, x̄, t) = µ−2/3 G(µ, µ1/3 x̄+ (µ − 1, 0, 0, µ − 1) , t),
(27)
Further, in order to understand the mutual relation between e and µ for which
we have interesting dynamical phenomena, it will be important to observe that after
rearranging and neglecting terms independent of x̄, (26) can be written as
H̄ e (µ, x̄, t) = H̄(µ, x̄) + eµ−1/3
2ȳ sin t − x̄ cos t
+ O(eµ2/3 ) + O(e2 µ−4/3 ),
3
r̄
(28)
on the following set
µ ∈ [0, 1],
r̄ > δ,
t ∈ R,
r̄ ≤ M1 ,
eµ−2/3 < κ
M1 µ1/3 < 1.
The Hamiltonian H̄ e generates a differential equation
x̄0 = f (µ, x̄) + eg(µ, x̄, t) + O(eµ1/3 ) + O(e2 µ−4/3 )
(29)
f (µ, x) = J∇H̄(µ, x)
(30)
g(µ, x, t) = J∇Ḡ(µ, x).
(31)
where
Remark 9 In our future consideration we shall use equation (28) in a neighborhood
of L̄µ2 = µ−1/3 (Lµ2 − (µ − 1, 0, 0, µ − 1)) of constant radius (later denoted as RHill ), in
which Lyapounov orbits reside. In such neighborhood the term O(eµ2/3 ) of (28) together
with higher order terms are uniform. It needs to be emphasized though that in order for
(28) to be valid for a given µ, we first need to choose e sufficiently small so that the
estimate eµ−2/3 < κ in (25) holds true.
3. From Lyapounov-Moser Theorem to Twist Property at Equilibrium
Points
In this section we shall show how one can prove the twist property at an equilibrium
point using the Lyapounov-Moser Theorem [23]. First the Theorem will be stated.
Next a number of observations on the Theorem in the special case of one real and one
pure imaginary eigenvalue with just two degrees of freedom will be made. This will be
followed by a brief outline of the construction by which the Theorem was proved [23]
from which the twist property will follow.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
12
Theorem 10 (The Lyapounov-Moser Theorem [23]) Let
q̇ν = Hpν (q, p)
(32)
ṗν = −Hqν (q, p)
ν = 1, . . . , n, be an analytic Hamiltonian system with n degrees of freedom and an
equilibrium solution q = p = 0. Let α1 , . . . , αn , −α1 , . . . , −αn be the eigenvalues of the
linearization of (32) at the equilibrium point. Assume that the eigenvalues
α1 , . . . , αn , −α1 , . . . , −αn
are 2n different complex numbers and that α1 , α2 are independent over the reals. Let us
also assume that for any integer numbers n1 and n2
αν 6= n1 α1 + n2 α2
for ν ≥ 3.
Then there exists a four parameter family of solutions of (32) of the form
qν = φν (ξ1 , ξ2 , η1 , η2 )
(33)
pν = ψν (ξ1 , ξ2 , η1 , η2 )
where
0 0
0 0
ξk (t) = ξk0 etak (ξ1 η1 ,ξ2 η2 ) ,
0 0
0 0
ηk (t) = ηk0 e−tak (ξ1 η1 ,ξ2 η2 )
for k = 1, 2,
(34)
and
a1 (ξ10 η10 , ξ20 η20 ) = α1 + ...,
a2 (ξ10 η10 , ξ20 η20 ) = α2 + ...
(35)
are convergent power series. The series φν , ψν converge in the neighborhood of the origin
and the rank of the matrix
Ã
!
φνξk φνηk
ψνξk ψνηk ν=1,2,...,n
k=1,2
is four. The solutions (33) depend on four small enough complex parameters ξk0 , ηk0 .
If in addition α1 , α2 , −α1 , −α2 contain their complex conjugates, the solution can
be chosen to be real, depending on 4 real parameters.
In the case of the PRC3BP, n is simply equal to two and the equations (33), (34)
describe all the solutions near the neighborhood of the equilibrium point. We will be
interested in the application of the Theorem to the libration point Lµ2 , where α1 is real
and α2 is pure imaginary. From now on we will restrict our discussion to this particular
case. The following remarks and lemmas adapt Theorem 10 to this setting.
Remark 11 When the system (32) is generated by a real Hamiltonian and if α1 is real
and α2 is pure imaginary then for the real solutions of (32) of the form (33) the functions
ξk (t) and ηk (t) are invariant under the involution [19, page 102]
Jw (ξ1 , ξ2 , η1 , η2 ) = (ξ¯1 , iη̄2 , η̄1 , iξ¯2 ).
(36)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
13
Let us also note that the original version of Theorem 10 in [23] contained an error.
There an involution
Jw (ξ1 , ξ2 , η1 , η2 ) = (ξ¯1 , η̄2 , η̄1 , ξ¯2 )
was proposed. This stands in conflict with a requirement that the transformation Φ in
the proof of Theorem 10 [23] should be canonical (See also equations (51) and (52)).
The reality condition (the fact that a point (ξ1 , ξ2 , η1 , η2 ) given in new coordinates
represents a point from R4 in original ones) is
Jw (ξ1 , ξ2 , η1 , η2 ) = (ξ1 , ξ2 , η1 , η2 ).
(37)
The subspace of C4 of fixed points of Jω , Fix(Jω ) is given by
ξ1 ∈ R, η1 ∈ R, ξ2 = reiϕ , η2 = ire−iϕ ,
where r, ϕ ∈ R. On Fix(Jw ) we will use the coordinates (ξ1 , η1 , r, ϕ).
Remark 12 From the proof of the convergence of the series (33), (35) during the proof
of Theorem 10 in [23], it follows that if we consider a family of Hamiltonians
Hλ : Rn × Rn → R,
which is an analytic function of all variables including the parameter and possesses for
each λ ∈ I, where I is a closed interval, a (locally) unique fixed point pλ of center-saddle
type depending analytically on λ, then the radius of convergence of the series (33), (35)
around pλ can be chosen uniformly for close values of λ.
Lemma 13 If α1 is real and α2 is pure imaginary then for all real solutions of (32) the
series a1 from the Theorem 10 is real and the series a2 is pure imaginary. Moreover if
we choose a real periodic solution
qν (t) = φν (0, ξ2 (t), 0, η2 (t))
pν (t) = ψν (0, ξ2 (t), 0, η2 (t))
ν = 1, 2
(38)
where ξ2 (t) and η2 (t) are given by (34), then there exist two real numbers r and ϕ such
that
ξ2 (t) = reta2 (0,ir
2 )+iϕ
η2 (t) = ire−ta2 (0,ir
2 )−iϕ
.
Proof. From Remark 11 we know that the real solutions satisfy the reality
condition (37). We therefore have
0 0
0 0
ξ10 eta1 (ξ1 η1 ,ξ2 η2 ) = ξ10 eta1 (ξ1 η1 ,ξ2 η2 )
³
´
0 0 0 0
0 0 0 0
ξ20 eta2 (ξ1 η1 ,ξ2 η2 ) = i η20 e−ta2 (ξ1 η1 ,ξ2 η2 )
0 0
0 0
0 0
0 0
η10 e−ta1 (ξ1 η1 ,ξ2 η2 ) = η10 e−ta1 (ξ1 η1 ,ξ2 η2 )
³
´
0 0 0 0
0 0 0 0
η20 e−ta2 (ξ1 η1 ,ξ2 η2 ) = i ξ20 eta2 (ξ1 η1 ,ξ2 η2 )
0 0
0 0
(39)
(40)
(41)
(42)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
14
if we choose t = 0 then from the above we can see that ξ10 and η10 are real and that
ξ20 = iη20 . Using the fact that ξ10 ,η10 ∈ R with (39) or (41) we can see that a1 must be
real. Using (40) or (42) and the fact that ξ20 = iη20 we can see that a2 is pure imaginary.
All periodic solutions have the initial conditions ξ10 = η10 = 0. If in addition we
choose ξ20 of the form ξ20 = reiϕ then for the solution to be real, from (40), we must have
ξ20 = iη20 . In such case equation (34) gives us the periodic solutions as
0 0
ξ2 (t) = ξ20 eta2 (0,ξ2 η2 ) = reta2 (0,ir
−ta2 (0,ξ20 η20 )
η2 (t) = η20 e
2 )+iϕ
= ire−ta2
,
(0,ir2 )−iϕ
(43)
.
Lemma 13 shows that all periodic solutions of (32) which are real and lie close to
the equilibrium point, are given by the equation
l(r, t) = (0, reta2 (0,ir
2 )+iϕ
, 0, ire−ta2 (0,ir
2 )−iϕ
),
(44)
when seen in the ξν , ην coordinates. Let us denote the set which contains these orbits
by
BR = {(0, reiϕ , 0, ire−iϕ )|ϕ ∈ [0, 2π), 0 ≤ r ≤ R},
(45)
where R is sufficiently small for the series a2 (0, ir2 ) to be convergent for r ≤ R.
Let P : R4 → R4 be the time 2π shift along the trajectory of (32) i.e.
P (q(t)) = q(t + 2π),
(46)
where q(t) is a solution of (32).
Lemma 14 If in the series a2 from Theorem 10 i.e.
a2 (ξ1 η1 , ξ2 η2 ) = α2 + a2,1 ξ1 η1 + a2,2 ξ2 η2 + ...
(47)
for the coefficient a2,2 ∈ R we have a2,2 6= 0, then for a sufficiently small R, the time 2π
shift along the trajectory P restricted to the set BR is an analytic twist map i.e.
P (r, ϕ) = (r, ϕ + f (r))
df
6= 0.
dr
(48)
Proof. In the ξ, η coordinates on BR from (44) we can see that the map P takes
form
´
¡
¢ ³
2
2
P 0, reiϕ , 0, ire−iϕ = 0, re2πa2 (0,ir )+iϕ , 0, ire−2πa2 (0,ir )−iϕ .
Keeping in mind that a2 is pure imaginary we can see that P (r, ϕ) = (r, ϕ−i2πa2 (0, ir2 )).
Since a2 (0, ir2 ) = α2 + a2,2 ir2 + O(r4 ) it is evident that if a2,2 6= 0, then for sufficiently
small r
¢
d ¡
a2 (0, ir2 ) 6= 0.
(49)
dr
Transition Tori in the Planar Restricted Elliptic Three Body Problem
15
Remark 15 Note that from Lemma 14 follows the twist property in action–angle
coordinates (I, ϕ) with I = r2 /2. This observation will play a role when applying the
KAM Theorem 29. Observe also that in action-angle coordinates the map P has the
following form P (I, ϕ) = (I, ϕ + 2πim(α2 ) + a2,2 I + O(I 2 )), which means that and the
ϕ
twist is more uniform and does not converge to zero as I → 0, because ∂P
(I, ϕ) → 2πa22
∂I
for I → 0.
We will now show how to determine whether for a given problem (32) we have
a2,2 6= 0. For this we will quickly outline the construction of Moser [23] in order to
obtain a formula for a2,2 . The construction is performed in the following two steps
Ψ
Φ
C4 → C4 → C4 ,
Ψ
Φ
(ξ1 , ξ2 , η1 , η2 ) 7→ (x1 , x2 , y1 , y2 ) 7→ (q1 , q2 , p1 , p2 ),
where the transformation Φ changes the system (32) in the q1 , q2 , p1 , p2 coordinates into
a system with a simplified form
ẋν = αν xν + fν (x, y)
ẏν = −αν yν + gν (x, y)
ν = 1, 2,
(50)
where α1 and α2 are the eigenvalues of the equilibrium point and f and g are power
series starting from quadratic terms. From the simplified form (50) the transformation
Ψ determines the series from Theorem 10.
The transformation Φ is a linear function which changes the coordinates so that
the linear part of the equations (32) in the new coordinates becomes generated by a
diagonal matrix. Moreover, the transformation Φ should be canonical i.e.
ΦT JΦ = J
where
Ã
J=
!
0 Id
−Id 0
(51)
Ã
and Id =
!
1 0
0 1
.
Moreover, Φ should satisfy the following reality condition [19, 23], which expresses the
fact that Jw is simply the map describing how the complex conjugation works in new
coordinates
Jz Φ = ΦJw ,
(52)
where
Jz (q1 , q2 , p1 , p2 ) = (q̄1 , q̄2 , p̄1 , p̄2 )
(53)
Jw (x1 , x2 , y1 , y2 ) = (x̄1 , iȳ2 , ȳ1 , ix̄2 ).
The construction of the transformation Ψ is done by comparison of coefficients. We
look for
Ψ = (φ1 (ξ, η), φ2 (ξ, η), ψ1 (ξ, η), ψ2 (ξ, η)),
Transition Tori in the Planar Restricted Elliptic Three Body Problem
with power series φν , ψν , av , ν = 1, 2 of the form
P
φν (ξ1 , ξ2 , η1 , η2 ) = 2k=1 δνk ξk + h.o.t.
P
ψν (ξ1 , ξ2 , η1 , η2 ) = 2k=1 δνk ηk + h.o.t.
16
(54)
such that
xν = φν (ξ1 , ξ2 , η1 , η2 )
(55)
yν = ψν (ξ1 , ξ2 , η1 , η2 ),
satisfy (50) if
ξ˙k = ak (ξ1 η1 , ξ2 η2 )ξk
(56)
η̇k = −ak (ξ1 η1 , ξ2 η2 )ηk .
To construct φν , ψν , aν one can rewrite using (55) and (56) the equation (50) as
³
´
P
∂φν
ν
ẋν = 2k=1 ∂φ
a
ξ
−
a
η
= αν φν + fν (φ, ψ)
k
k
k
k
∂ξk
∂ηk
´
ν = 1, 2.
P2 ³ ∂ψν
ν
a η = −αν ψν + gν (φ, ψ),
ẏν = k=1 ∂ξk ak ξk − ∂ψ
∂ηk k k
(57)
and compare the coefficients in (57). Let us denote by φν,N , ψν,N , aν,N the coefficients in
the series φν , ψν , aν which come from the homogenous polynomials of order N . We can
rewrite the part of (57) which contains all the terms of order N as
´
³
P2
∂
∂
k=1 αk ξk ∂ξk − ηk ∂ηk φν,N + . . . + δνk ξk ak,N −1 = αν φν,N + . . .
³
´
(58)
P2
∂
∂
α
ξ
ψ
+
.
.
.
−
δ
η
a
=
−α
ψ
+
.
.
.
−
η
k
k
ν,N
νk
k
k,N
−1
ν
ν,N
k
k=1
∂ξk
∂ηk
where the dots indicate all the terms which can be computed from φν,l , ψν,l , aν,l−1 with
l = 1, . . . , N − 1.
The nature of equations (58) suggests that the series can be constructed by
induction starting with the lowest terms. It turns out though that not all of the
coefficients can be computed from (58). This is because some of the terms in (58)
cancel each other out. If we consider a homogenous polynomial cξ1n1 η1m1 ξ2n2 η2m2 of order
N from φν,N , such term will cancel out in (58) if
2
X
k=1
µ
αk
∂
∂
− ηk
ξk
∂ξk
∂ηk
¶
cξ1n1 η1m1 ξ2n2 η2m2 − αν cξ1n1 η1m1 ξ2n2 η2m2 = 0.
This can happen only if we have
2
X
αk (nk − mk ) − αν = 0.
(59)
k=1
By the assumption of the Theorem 10 that for any t ∈ R we have tα1 + α2 6= 0, we can
see that (59) is true only for the terms of the form cξv (ξ1 η1 )n1 (ξ2 η2 )n2 . The value of the
Transition Tori in the Planar Restricted Elliptic Three Body Problem
17
coefficient c corresponding to such terms is chosen from an appropriate normalization
condition [23]. In our case though the choice of the normalization does not play an
important role. We are interested in computation of the term a2,2 and this can be done
by induction starting with N = 1 and stopping at N = 3. For N = 1 all the terms are
uniquely determined. For N = 2 there are no terms which would cancel out in (58).
For N = 3 we can use the first equation from (58) to find a2,2 . There the coefficient a2,2
stands together with ξ2 (ξ2 η2 ) and the term for ξ2 (ξ2 η2 ) in φν,3 will cancel out from the
equation. Therefore a2,2 is uniquely determined since it depends only on φν,1 , ψν,1 , φν,2
and ψν,2 . Using this procedure we can obtain a direct formula for the term a2,2 .
Lemma 16 If
P
ν
xi1 xj2 y1k y2l
fν (x1 , x2 , y1 , y2 ) = i,j,k,l≥1 fijkl
P
ν
gν (x1 , x2 , y1 , y2 ) = i,j,k,l≥1 gijkl
xi1 xj2 y1k y2l
ν = 1, 2.
then
a2,2 =
1
2
1
2
1
2
1
2
2
(−f1,1,0,0
f0,1,0,1
− f0,1,1,0
g0,1,0,1
+ f0,0,1,1
g0,2,0,0
− f0,2,0,0
f0,1,0,1
α2
2
2
2
1
2
2
2
+ 2g0,2,0,0
f0,0,0,2
+ f1,0,0,1
f0,2,0,0
− g0,1,0,1
f0,1,0,1
) + f0,2,0,1
(60)
Proof. The above can be checked from the formula (58) by direct computation.
This ends our construction of the coefficient a2,2 .
Let us now briefly turn to the relation between the energy and the radius r of the
periodic orbits (44).
Lemma 17 For sufficiently small r the energy of the orbit lr (44) i.e.
h(r) = H(Φ (Ψ(lr ))),
(61)
is equal to
1
h(r) = H(0) + D2 H(0) (Φ(0, 1, 0, i)) r2 + o(r2 ),
(62)
2
where D2 H(0) (Φ(0, 1, 0, i)) denote the value of the quadratic form D2 H(0) on the vector
Φ(0, 1, 0, i).
Proof. Since the problem (32) is autonomous the energy is constant along the
orbit lr . Without any loss of generality we can therefore assume that ϕ in equation (44)
for lr is zero and compute
h(r) = H(Φ (Ψ(l(r, 0)))).
Let us first note that the construction of Ψ = (φ1 , φ2 , ψ1 , ψ2 ) produced power series of
the form (54), hence
Ψ(l(r, 0)) = (φ1 , φ2 , ψ1 , ψ2 )(l(r, 0)) = (0, r, 0, ir) + O(r2 ).
The transformation Φ is linear and therefore
Φ (Ψ(l(r, 0))) = rΦ(0, 1, 0, i) + O(r2 ).
(63)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
18
We can compute h(r) as
h(r) = H(Φ (Ψ(l(r, 0))))
(64)
= H(0) + DH(0) (Φ (Ψ(l(r, 0))))
1
+ D2 H(0) (Φ (Ψ(l(r, 0))))2 + o(|Φ (Ψ(l(r, 0)))|2 ).
2
Since zero is an equilibrium point we know that DH(0) = 0, thus by substituting (63)
into (64) we obtain our claim.
4. Twist in the PRC3BP at Lµ2
As mentioned in Section 2, we have a family of periodic Lyapounov orbits lµ (C) around
Lµ2 for energies larger than and sufficiently close to the energy C2µ of the libration point
Lµ2 . This family of orbits corresponds to the set BR (see (45)) of orbits constructed in
the previous section. In this section we will show that for sufficiently small µ we have
a twist property on this family of periodic orbits. The main idea for the proof is to
approximate the PRC3BP (with Hamiltonian (16)) expressed in Hill’ coordinates (15)
with the Hill’s problem (18). First we shall consider the Hill’s problem (18) where we
Hill
have explicit formulas for the libration point LHill
2 , the linearized vector field at L2
its eigenvalues etc., which will allow us to compute the twist coefficient aHill
2,2 . We will
µ
then show that the coefficients a2,2 computed for the PRC3BP in Hill’s coordinates,
converges to aHill
2,2 as µ tends to zero.
Lemma 18 Let P Hill be the time 2π shift along the trajectory Poincaré map of the Hill’s
problem (18). Then there exists a radius RHill ∈ R, such that the map P Hill expressed
in in radius angle coordinates on the set of Lyapounov orbits around LHill
2
P Hill : [0, RHill ] × S 1 → [0, RHill ] × S 1 ,
is a twist map.
Proof. We will apply the procedure from the previous section and compute aHill
2,2
−1/3
−1/3
Hill
Hill
, 0, 0, 3 p
). The linear terms ofp
(18) in L2 are
for the equilibrium point L2 = (3
√
√
given by (20) with eigenvalues ±α1 , ±α2 , α1 = 1 + 2 7 and α2 = 1 − 2 7. The
first of the two is real and the second is pure imaginary. We will choose the function
ΦHill composed of the eigenvectors of the eigenvalues ±α1 and ±α2


λ1
β
λ2
−iβ̄
 −9λ1 √1
−β9 α √17−4
9λ2 α √17+4
−iβ̄9 α √17−4 

α1 ( 7+4)
)
)
) 
2(
1(
2(
Hill
√
√
√
√
,
(65)
Φ =
7+3
7−3
7+3
7−3
 9λ1 √

√
√
√
−β9
−9λ
−i
β̄9
2

α1 ( 7+4)
α2 ( 7−4)
α1 ( 7+4)
α2 ( 7−4) 
2
2
−λ1 3+2√7
β √7−3
−λ2 3+2√7
−iβ̄ √7−3
Transition Tori in the Planar Restricted Elliptic Three Body Problem
19
with λ1 , λ2 ∈ R, β ∈ C. The above transformation ΦHill satisfies the reality condition
(52), and if we choose the coefficients λ1 , λ2 , β as
s ¡√
¢
7+4
1 α1
√
λ1 = −λ2 =
,
(66)
6
7
s
¡√
¢
7−4
1 iα2
√
β=
,
6
7
then ΦHill is also canonical. Computing the power series fν and gν from Lemma 16
at LHill
and using (60) to compute the term aHill
2
2,2 , by rather laborious computations
(performed in Maple) one will obtain
√
3
´
9³ √
Hill
a2,2 =
102 7 − 57 ≈ 1. 976 7.
224
Since aHill
2,2 6= 0, by Lemma 14 we have the twist property in the radius angle coordinates
for all r such that 0 < r < RHill , where RHill is sufficiently small.
Remark 19 Let us stress that RHill is independent of µ. From our attempts of rigorous
estimation of RHill , following the estimates conducted during the proof of Theorem 10
in [23], we obtain that that RHill ≈ 10−4 .
One can also apply the procedure outlined in Section 3 to compute the coefficient
for the PRC3BP with Hamiltonian (16) expressed in Hill’ coordinates (15). To
do so one needs to compute the libration point L̄µ2 = µ−1/3 (Lµ2 − (µ − 1, 0, 0, µ − 1)),
compute the expansion of the vector field at L̄µ2 k up to the order three, compute the
linear change of coordinates Φ = Φ(µ) and compute aµ2,2 using Lemma 16. Numerical
results of such computations for a selection of parameters from the family {µk } from
Theorem 3 are given in the below table. The values µk chosen in the table are the
numerical approximations of the series (11) from [20].
aµ2,2
Transition Tori in the Planar Restricted Elliptic Three Body Problem
k
µk
2 0.4253863522E-2
3 0.6752539971E-3
4 0.2192936884E-3
20
aµ2,2k
1.967649155
1.968237635
1.970039000
10
11
12
0.92907436E-5
0.68212830E-5
0.51549632E-5
1.973971883
1.974219993
1.974426964
50
60
70
0.582146E-7
0.336890E-7
0.212152E-7
1.976164111
1.976252225
1.976315175
200
0.9096E-9
1.976563023
aHill
2,2 ≈ 1.9767
Table 1. The twist coefficient a2,2 for various masses µk .
Theorem 20 Let P µ denote the the time 2π shift along the trajectory P µ of the
PRC3BP in Hill’s coordinates (15). Then for sufficiently small µ the map P µ
expressed in in radius angle coordinates on the set of Lyapounov orbits around L̄µ2 =
µ−1/3 (Lµ2 − (µ − 1, 0, 0, µ − 1)) is an analytic twist map i.e. for r < RHill
P µ (r, ϕ) = (r, ϕ + f (r))
and
df
6= 0
dr
for all r ∈ [0, RHill ].
Proof. We observe that L̄µ2 depends analytically on µ1/3 . First let us note that
since the operator Φ from our construction brings the derivative of the vector field to
the Jordan form, the operator Φµ3Body for the PRC3BP can be chosen close (depending
analytically on µ1/3 ) to the operator ΦHill . The same can be said about the coefficients fν
and gν from Lemma 16, since those come from the Taylor expansion up to the third order
of the vector field at L̄µ2 . Moreover, from the proof of the Lyapounov Moser Theorem 10
in [23], we know that the radius of convergence of the series from the Theorem 10 can
be chosen to be independent from µ, for µ sufficiently close to zero. This means that
the coefficient aµ2,2 constructed for he PRC3BP will tend to the coefficient aHill
2,2 of the
Hill’s problem
lim aµ2,2 = aHill
(67)
2,2 ≈ 1. 976 7.
µ→0
Transition Tori in the Planar Restricted Elliptic Three Body Problem
21
Remark 21 From Theorem 20 and the results from Table 1, it is reasonable to believe
that we will have twist for all µk for k ≥ 2. Let us point out that in the Hill’s coordinates
the radius of convergence can be chosen to be independent from µ. In the original
coordinates of the PRC3BP though, since rHill = µ1/3 r, this radius will depend and
decrease with µ as Rµ = µ1/3 RHill . This means that in the original coordinates we will
have the twist property only for orbits with a radius smaller than µ1/3 RHill .
5. Normal hyperbolicity, KAM theorem and the persistence of Lyapounov
orbits
In this Section we briefly recall some facts from the normal hyperbolicity theory and
a version of the K.A.M. (Kolmogorov, Arnold, Moser) Theorem. We then apply the
results to obtain the persistence result of a Cantor family of Lyapounov orbits around
Lµ2 for the perturbation from PRC3BP to PRE3BP. Our approach closely follows the
method of [8]. We will therefore rewrite the theorems used in [8] and verify that their
assumptions are satisfied in our particular setting.
First let us recall the results concerning normal hyperbolicity.
Definition 22 ([8, A1]) Let M be a manifold in Rn and Φt a C r , r ≥ 1 flow on it.
We say that a (smooth) manifold Λ ⊂ M – possibly with boundary – invariant under Φt
is α-β normally hyperbolic when there is a bundle decomposition
T M = T Λ ⊕ E s ⊕ E u,
invariant under the flow, and numbers C > 0, 0 < β < α, such that for x ∈ Λ
v ∈ Exs ⇔ |DΦt (x)v| ≤ Ce−αt |v| ∀t > 0,
(68)
v ∈ Exu ⇔ |DΦt (x)v| ≤ Ceαt |v| ∀t < 0,
(69)
v ∈ Tx Λ ⇔ |DΦt (x)v| ≤ Ceβ|t| |v| ∀t.
(70)
Theorem 23 ([8, A7]) Let Λ be a compact α-β normally hyperbolic manifold (possibly
with a boundary) for the C r flow Φt , satisfying the Definition 22. Then there exists a
sufficiently small neighborhood U of Λ and a sufficiently small δ > 0 such that
(i) The manifold Λ is C min(r,r1 −δ) , where r1 = α/β.
(ii) For any x in Λ, the set
Wxs = {y ∈ U : dist(Φt (y), Φt (x)) ≤ Ce(−α+δ)t for t > 0}
= {y ∈ U : dist(Φt (y), Φt (x)) ≤ Ce(−β−δ)t for t > 0}
is a C r manifold and Tx Wxs = Exs .
(iii) The bundles Exs are C min(r,r0 −δ) in x, where r0 = (α − β)/β, and
WΛs = {y ∈ U : dist(Φt (y), Λ) ≤ Ce(−α+δ)t for t > 0}
= {y ∈ U : dist(Φt (y), Λ) ≤ Ce(−β−δ)t for t > 0}
Transition Tori in the Planar Restricted Elliptic Three Body Problem
22
is a C min(r,r0 −δ) manifold. Moreover Tx WΛs = Exs . Finally
[
WΛs =
Wxs .
x∈Λ
Moreover, we can find a ρ > 0 sufficiently small and a C min(r,r0 −δ) diffeomorphism
from the bundle of balls of radius ρ in EΛs to WΛs ∩ U.
Remark 24 An analogous theorem can be stated for WΛu by considering the flow Φ−t .
The following Theorem and two Remarks concern the persistence of the normally
hyperbolic manifold and its stable and unstable manifolds.
Theorem 25 ([8, A.14]) Let Λ ⊂ M (Λ not necessarily compact) be α-β normally
hyperbolic for the flow Φt generated by the vector field X, which is uniformly C r in a
neighborhood U of Λ such that dist(M \ U, Λ) > 0. Let Ψt be the flow generated by
another vector field Y which is C r and sufficiently C 1 close to X. Then we can find a
manifold Γ which is α0 -β 0 normally hyperbolic for Y and C min(r,r1 −δ) close to Λ, where
r1 = α/β.
The constants α0 , β 0 are arbitrarily close to α, β if Y is sufficiently C 1 close to X.
The manifold Γ is the only C min(r,r1 −δ) normally hyperbolic manifold C 0 close to Λ
and locally invariant under the flow of Y.
The above Theorem is extended to give us a smooth dependence on the parameter
by the following two remarks.
Remark 26 (see [8, observation 1. page 390]) Assume that we have a family of
flows Φt,e , generated by vector fields Xe which are jointly C r in all its variables (the
base point x and the parameter e). Let Λe be the normally hyperbolic manifold Γ from
Theorem 25 for the flow Φt,e . Then there exists a C min(r,r1 −δ) mapping F : Λ × I → M,
where r1 = α/β and I ⊂ R is an interval containing zero, such that F (Λ, e) = Λe and
F (·, 0) is the identity.
Remark 27 (see [8, observation 2. page 390]) For a family of flows Φt,e with the
same assumptions as in Remark 26, there exists a C min(r,r1 −δ) ( r1 = α/β ) mapping Rs :
s
WΛs × I → M such that Rs (WΛs , e) = WΛ,e
, Rs (·, e)|Λ = F (·, e), Rs (Wxs , e) = WFs (x,e),e .
An analogous mapping Ru also exists for WΛu .
Let us now turn to a quantitative version of the KAM Theorem used in [8]. Let us
recall that a real number ω is called a Diophantine number of exponent θ if there exists
a constant C > 0 such that
¯
¯
¯
¯
¯ω − p ¯ ≥ C
¯
q ¯ q θ+1
for all p ∈ Z, q ∈ N.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
23
Definition 28 Let (M, ωM ), (N, ωN ) be two symplectic manifolds of same dimension.
If (M, ωM ), (N, ωN ) are exact symplectic (i.e. there exist one-forms αM , αN such that
ωM = dαM , ωN = dαN ) then we say that a diffeomorphism
F :M →N
is exact symplectic when there exists a real valued function G on M such that
F ∗ αN − αM = dG.
Theorem 29 (KAM Theorem [8, Theorem 4.8]) Let f : [0, 1] × T1 → [0, 1] × T1
be an exact symplectic C l map with l ≥ 6.
Assume that f = f0 + ef1 , where e ∈ R,
f0 (I, ϕ) = (I, ϕ + A (I)) ,
(71)
¯ ¯
¯ ≥ M , and kf1 k l ≤ 1.
A is C l , ¯ dA
C
dI
Then, if e1/2 M −1 = ρ is sufficiently small, for a set of Diophantine numbers σ of
exponent θ = 5/4, we can find invariant tori which are the graph of C l−3 functions uσ ,
the motion on them is C l−3 conjugate to the rotation by σ and the tori cover the whole
¡
¢
annulus except a set of measure smaller than O M −1 e1/2 .
Moreover, we can find expansions
uσ = u0σ + eu1σ + rσ ,
(72)
with u0σ = A−1 (σ), krσ kC l−4 ≤ O (e2 ) , and ku1σ kC l−4 ≤ O(1).
All of the above results have been taken from [8]. Now we will apply them to the
setting of the PRE3BP. We will first show that the set of the Lyapounov orbits of the
PRC3BP is normally hyperbolic.
Let φet,s : R4 → R4 be given by
φet,s (x) = q(s + t),
where q(·) is the solution for the PRE3BP in Hill’s coordinates (15) (generated by the
Hamiltonian (16)), with an initial condition q(s) = x̄. We will define the flow on the
extended phase space Φet : R4 × R → R4 × R as
Φet (x̄, s) = (φet,s (x̄), s + t).
(73)
Observe that the flow Φet is 2π periodic with respect to s variable, hence may be
equivalently treated as a flow on R4 × S 1 . This will later give us uniform C r estimates.
We shall use a notation
l(r) = l(r, S 1 ) × S 1
to denote a torus of all trajectories of Lyapounov orbits l(r, t) (see (44)) of radius r in
the extended phase space.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
24
Lemma 30 For a sufficiently small mass µ the set
Λ = {l(r)|r ∈ [0, RHill )}
of Lyapounov orbits of the PRC3BP (in the extended phase space) is α-β normally
hyperbolic, where α > 0 is close to the real eigenvalue α1 at the Libration point
L̄µ2 = µ−1/3 (Lµ2 − (µ − 1, 0, 0, µ − 1)) and β > 0 can be chosen arbitrarily close to zero.
Proof. Consider the ξ1 , ξ2 , η1 , η2 coordinates from the previous section together
with a time t coordinate. The ξ1 , η1 are the coordinates of the hyperbolic expansion and
ξ1 , η1 are the coordinates of the twist rotation around the libration point L̄µ2 . We have
M = {(ξ1 , ξ2 , η1 , η2 , t)|ξ1 , η1 ∈ R, ξ2 = reiϕ , η2 = ire−iϕ ,
r ∈ [0, RHill ), ϕ ∈ [0, 2π), t ∈ [0, 2π)}.
We can define
E u = {(ξ1 , 0, 0, 0, 0)|ξ1 ∈ R},
E s = {(0, 0, η1 , 0, 0)|η1 ∈ R},
T Λ = {(0, ξ2 , 0, η2 , t)|ξ2 = reiϕ , η2 = ire−iϕ , r ∈ R,ϕ ∈ [0, 2π), t ∈ [0, 2π)},
then we will have T M = E u ⊕ E s ⊕ T Λ. The conditions (68) and (69) are satisfied with
a coefficient α > 0 close to the eigenvalue α1 at L̄µ2 because the coordinates ξ1 and η1
are the coordinates of hyperbolic expansion
and contraction. For sufficiently small µ
p
√
Hill
the eigenvalue α1 is close to α1 = 1 + 2 7 ≈ 2. 508 3 of the Hill’s problem.
Let Φt be the flow in the extended phase space. From (44) for x = (0, ξ2 , 0, η2 , t) ∈ Λ
with ξ2 = reiϕ , η2 = ire−iϕ we have
¯
¯
¯d
¯
2 ¯
¯
kDΦt (x)k ≤ 1 + r ¯ a2 (0, ir )¯ t,
dr
where a2 is the function given by (35) in Theorem 10. The growth of derivative of
DΦt (x) is at most linear in t. For any β > 0 there exists a constant C > 0, such that
for all v ∈ Tx Λ and all t
|DΦt (x)v| ≤ Ceβ|t| |v|.
(74)
We will now define the time 2π shift along a trajectory Poincaré map and later
apply the KAM Theorem to it. By Lemma 30 for e = 0 we have an α-β normally
hyperbolic invariant manifold for Φ0t of the form Λ = {l(r)|r ∈ [0, RHill )}. Let U be an
open neighborhood of Λ. We will define time 2π shift along a trajectory Poincaré map
Pte0 : U ∩ {t = t0 } → R4 as
Pte0 (x) = φe2π,t0 (x).
We are now ready to apply Theorems 25 and 29 to obtain the following persistence
result for the family of the Lyapounov orbits.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
25
Theorem 31 Let R < RHill be a fixed number and κ be the parameter from (25). If we
choose sufficiently small µ∗ > 0 then for all µ > 0 and e > 0 for which eµ−2/3 < κ the
normally hyperbolic manifold (with a boundary; considered in the extended phase space)
Λ = {l(r)|r ∈ [0, R)} of the PRC3BP persists under the perturbation to PRE3BP with
the parameter e, to a normally hyperbolic manifold (with a boundary) Λe . Moreover, for
any such e there exists a Cantor set C ⊂ [0, R] such that for any r ∈ C there exists an
invariant (two dimensional) torus
©
ª
le (r) = lte0 (r) |t0 ∈ S 1 ,
where lte0 (r) is an one–dimensional torus invariant under the map Pte0 . The family of
¡
¢1/2
tori le (r) for r ∈ C, covers Λe except a set of a measure smaller than O( eµ−1/3
).
Proof. The PRE3BP in Hill’s coordinates (15) is generated by the Hamiltonian H̄ e
from (28). In the neighborhood U of Λ the r̄ from (17) and (28) is bounded, which means
that the PRE3BP is a uniform perturbation of the Hill’s problem (18). By Lemma 30
we know that Λ is normally hyperbolic for the PRC3BP. Applying Theorem 25 and
Remark 26 we obtain a family of normally hyperbolic manifolds Λe locally invariant
under Φet , and a function F : Λ × [0, e0 (µ)] → R4 × S 1 such that
F (Λ, e) = Λe = {(Λt,e , t) |Λt,e ⊂ R4 , t ∈ S 1 }.
From the Implicit Function Theorem we know that the libration point L̄µ2 continues
for small values of e to a 2π periodic orbit L̄µ,e
We can modify F so that
2 (t).
µ
µ,e
r1
F (L̄2 , t, e) = L̄2 (t). By Remark 26 the function F is C , where r1 = p
α/β. From the
√
proof of Lemma 30 we know that for sufficiently small µ we have α ≈ 1 + 2 7 and
that β > 0 can be chosen to be arbitrarily close to zero. This means that for sufficiently
small µ, the function F is C k for any given k > 0. Since Λe is locally invariant under Φet
and the flow is 2π periodic, for any t0 ∈ S 1 the manifold Λt0 ,e is locally invariant under
Pte0 .
Let us fix t0 = 0, fix small µ > 0 and fix a Poincaré map P e := Pte0 =0 (here we
could consider a map Pte0 for any t0 ∈ S 1 , but we fix t0 = 0 for simplicity) and consider
e such that eµ−2/3 < κ, which ensures that (28) is valid. We shall use a notation
BR = π{t=0} {l(r)|r ≤ R} for the set of Lyapounov orbits with radius smaller or equal
to R. We will now show that for sufficiently small eµ−1/3 the Poincaré map
P e : F (BR , 0, e) → Λ0,e
(75)
is properly defined and symplectic. By (28), for sufficiently small eµ−1/3 we can see that
(75) is properly defined because Λ0,e is locally invariant. The map P e is a restriction
to Λ0,e of a time 2π shift along a trajectory of a Hamiltonian system. Such shift is
symplectic for the standard form ω = dx̄ ∧ dp̄x + dȳ ∧ dp̄y (here we use notation
x̄ = (x̄, ȳ, p̄x , p̄y ) for coordinates since we are working in Hill’s coordinates (15)). In
order to show that P e is symplectic it is therefore sufficient to show that ω is non
Transition Tori in the Planar Restricted Elliptic Three Body Problem
26
degenerate on Λ0,e . For sufficiently small µ and e the manifold Λ0,e is arbitrarily close
to the manifold of the Lyapounov orbits of the Hill’s problem (18), which in turn,
for r sufficiently close to zero, is arbitrarily close to the vector
space V given by the
p
√
Hill
eigenvectors of the pure complex eigenvalues ±α2 = ± 1 − 2 7. To show that ω
is not degenerate on Λ0,e it is therefore sufficient to show that ω is not degenerate
on V. The eigenvectors corresponding to ±α2Hill are v and −iv̄, where v is the second
column in ΦHill (see (65)), which was symplectic, therefore ω(v, −iv̄) = 1. The space
V is spanned by x1 and x2 , where v = x1 + ix2 . An easy computation shows that
1 = ω(v, −iv̄) = −2ω(x1 , x2 ), which means that ω is not degenerate on V .
Now we will use the KAM Theorem 29 to show that most of the Lyapounov orbits
on Λ0,e survive under a sufficiently small perturbation. Let us first note that even though
the Theorem is stated for a map f on [0, 1] × T1 , the KAM result is local by its nature
and also holds for a map f : [0, 1] × T1 → R × T1 , as will be the case in our setting.
Let ω denote the standard symplectic form in R4 i.e. ω = dx̄ ∧ dp̄x + dȳ ∧ dp̄y . Let ω e
denote the induced form on Λ0,e . There exist C r1 −2 (jointly with the parameter e) close
to identity coordinate maps ce : Λ0,e → Λ0,0 which transport the symplectic forms ω e
into the standard one (see [8, page 367]). The map
P̄ e := ce ◦ P e ◦ (ce )−1 : BR → BRHill
is properly defined for sufficiently small e. Clearly for e = 0 we have P̄ 0 = P 0 . From
the fact that P e is symplectic and the fact that P 0 = Pte=0
is a twist map follow the
0 =0
e
0
same properties for our maps P̄ and P̄ respectively. Now we pass to the action angle
coordinates. From Lemma 14 we have that P̄ 0 has the form (71). Exact simplecticity of
P̄ e is a direct consequence of simplecticity combined with invariance of the origin. To
apply the KAM Theorem 29 what is now left is to show that
° e
°
°P̄ − P̄ 0 ° r −2 = O(eµ−1/3 ).
(76)
C 1
This comes from the fact that in the neighborhood of L̄µ2 the perturbing term in (28)
is uniformly O(eµ−1/3 ) in the C l norm. Observe that this estimate holds both in the
original coordinates and in the action angle coordinates, because the origin is the fixed
point for P̄ e . Therefore the time 2π shift maps Pte0 and Pt00 are also O(eµ−1/3 ) close,
from which (76) follows. This gives us a Cantor family of invariant tori l0e (r) for r ∈ C.
Now for r ∈ C we can define
le (r) := {Φet (x, 0)|x ∈ l0e (r), t ∈ [0, 2π)}
lte0 (r) := Φet0 (l0e (r) , 0).
¢1/2
¡
) follows from
The fact that the complement of the Cantor set C is O( eµ−1/3
the KAM Theorem (see also [24] for more details).
In the above argument we require that eµ−1/3 < c with some sufficiently small c
(which is independent of µ) so that both the normally hyperbolic theorem and KAM
can be applied. Let finish by observing that for any given c, by choosing sufficiently
Transition Tori in the Planar Restricted Elliptic Three Body Problem
27
small µ∗ > 0 and requiring that µ < µ∗ and eµ−2/3 < κ the estimate eµ−1/3 < c follows.
Remark 32 An identical argument to the above proof can be performed to obtain a
mirror result to Theorem 31 for any fixed parameter µk . To do so though one would
have to verify the twist condition. This has been verified numerically for a sequence of
parameters in Table 1. It is visible that as the masses µk decrease the twist coefficient
increases to infinity. It is therefore reasonable to believe that the twist condition holds
for all parameters µk . Let us emphasize that for sufficiently large k we have rigorously
verified the twist condition in Lemma 20, hence the claim of Theorem 31 holds for µk
sufficiently large k.
Remark 33 We can set R from Theorem 31 to be the radius of one of the surviving
tori. This will ensure that Λe is an invariant manifold with a boundary le (R). We should
also emphasize that the choice of R is independent of µ.
If for two surviving tori le (r1 ) and le (r2 ) with r1 < r2 there does not exist an
r ∈ (r1 , r2 ) for which the torus l(r) is perturbed to an invariant torus of the PRE3BP,
then we say that there exists a gap between the tori le (r1 ) and le (r2 ).
Proposition 34 Let eµ−1/3 be sufficiently small so that the claim of Theorem 31 holds.
¡
¢1/2
Then there exists an interval I ⊂ [0, R] with measure of order eµ−1/3
, for which the
set I ∩ C for which Lyapounov orbits persist under perturbation has gaps smaller than
ζeµ−1/3 , where ζ > 0 is any given constant.
Proof. The fact that such an interval exists will follow from the fact that the
¡
¢1/2
complement of the Cantor set C is of the measure O( eµ−1/3
). Let us divide the
interval [0, R] into n equal parts. If on every interval the set C contains gaps larger than
¡
¢1/2
ζeµ−1/3 , then from the fact that the measure of the complement of C is O( eµ−1/3
)
¡ −1/3 ¢1/2
¡ −1/3 ¢1/2
(let us say that this O( eµ
) is equal to M eµ
for some M > 0) the
number of such intervals n must satisfy
¡
¢1/2
nζeµ−1/3 ≤ M eµ−1/3
,
¢−1/2
¡
. If we divide the interval [0, R] into a slightly
which means that n ≤ ζ1 M eµ−1/3
larger number ñ of equal intervals then at least one of them (this will be our interval I)
cannot contain a gap larger than ζeµ−1/3 . The size of such an interval is equal to
¡
¢1/2
1
R
≈
Rζ eµ−1/3
.
ñ
M
Transition Tori in the Planar Restricted Elliptic Three Body Problem
28
6. Melnikov method
In the previous section we have shown that the normally hyperbolic manifold with
a boundary Λ = {l(r) : r ∈ [0, R]} (considered in the extended phase space) of the
PRC3BP (28) in Hill’s coordinates (16) persists under perturbation to Λe which
is a normally hyperbolic invariant manifold with a boundary of the PRE3BP with
eccentricity e. Moreover, we have shown that Λe contains a Cantor set of two dimensional
invariant KAM tori. In this section we will consider the problem of intersections of the
stable and unstable manifolds of such tori.
In this section we shall once again work in the Hill’s coordinates (15) and
Hamiltonian (26). It will also be convenient for us to parameterize the manifolds Λ
and Λe using the radius angle coordinates r, ϕ of the Lyapounov orbits from Section
3 together with time t ∈ S 1 . For e = 0 we thus use a natural parameterization of
the Lyapounov orbits by their Birkhoff normal form coordinates (coordinates obtained
from Theorem 10). After the perturbation it will be enough for us to use the fact that
the parametrization is smooth (in fact, from the proof of Theorem 31 we know that it
will be C r1 with r1 = α/β) and that we can parametrize (see proof of Theorem 31) the
perturbed libration point L̄µ2 = µ−1/3 (Lµ2 − (µ − 1, 0, 0, µ − 1)) (which continues to a 2π
periodic orbit) by r = 0. We can not assume though that the surviving perturbed KAM
tori are parameterized by r. Our parametrization simply follows from the Normally
Hyperbolic Invariant Manifold Theorem (see Theorem 25 and Remarks 26, 27) without
involving the KAM Theorem.
Let us recall that for µ close to µk from Theorem 3 prior to the perturbation the
fibres Wps for p ∈ Λ intersect transversally with the section {ȳ = 0} (see Section 2,
Theorems 3, 4 and also [20]). The same goes for the unstable fibers Wpu . This means
that for sufficiently small 0 < e the same will hold for the stable fibres Wps,e and unstable
fibres Wpu,e of points p ∈ Λe . Each point p ∈ Λe can be parameterized by µ, r, ϕ and
t0 . The fibres of such points Wpu,e , Wps,e are one dimensional and contained in sections
Σt0 = {(q, t0 )|q ∈ R4 }. The intersections of Wpu,e and of Wps,e with {ȳ = 0} are functions
of (µ, r, ϕ, t0 , e). For a point p ∈ Λe parameterized by µ, r, ϕ and t0 we introduce the
following notation for the first intersections of Wps,e and of Wpu,e with {ȳ = 0}
(ps (r, ϕ, t0 , e), t0 ) = Wps,e ∩ {y = 0},
(pu (r, ϕ, t0 , e), t0 ) = Wpu,e ∩ {y = 0}.
Let q s (r, ϕ, t0 , e, t) and q u (r, ϕ, t0 , e, t) be the orbits (considered in the standard (not
extended) phase space) of the PRE3BP, which start from the points ps (r, ϕ, t0 , e) and
pu (r, ϕ, t0 , e) respectively at time t = t0 i.e.
q s (r, ϕ, t0 , e, t0 ) = ps (r, ϕ, t0 , e),
q u (r, ϕ, t0 , e, t0 ) = pu (r, ϕ, t0 , e).
(ps , pu , q s , q u depend on the choice of µ, but we omit this in our notations for simplicity).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
29
Let us note that for parameters µ = µk from Theorem 3
q s (0, ϕ, t0 , 0, t) = q u (0, ϕ, t0 , 0, t)
(77)
= q̄µ0 k (t − t0 )
¡
¢
= µ−1/3 qµ0 k (t − t0 ) − (µk − 1, 0, 0, µk − 1) ,
where qµ0 k (t) is the homoclinic orbit to Lµ2 k in the PRC3BP defined just after the
statement of Theorem 3. Let us remind the reader that qµ0 k (0) ∈ {y = 0}.
Lemma 35 For fixed µ and i ∈ {s, u}
q i (r, ϕ, t0 , 0, t0 ) = q i (0, ϕ, t0 , 0, t0 ) + O(r),
q i (r, ϕ, t0 , e, t0 ) = q i (r, ϕ, t0 , 0, t0 ) + e
∂q i
(r, ϕ, t0 , 0, t0 ) + o(e),
∂e
∂q i
∂q i
(r, ϕ, t0 , 0, t0 ) =
(0, ϕ, t0 , 0, t0 ) + O(r),
∂e
∂e
where the bounds o(e) and O(r) are independent from t0 .
In addition for f, g from (29)
µ
¶
¡
¢ ∂q i
d ∂q i
(0, ϕ, t0 , 0, t) = Df µ, q i (0, ϕ, t0 , 0, t)
(0, ϕ, t0 , 0, t)
dt ∂e
∂e
¡
¢
+ g q i (0, ϕ, t0 , 0, t), t ,
and
∂q i
(0, ϕ, t0 , 0, t)
∂e
(78)
(79)
(80)
(81)
is bounded for all t ∈ [t0 , +∞) for i = s (or t ∈ (−∞, t0 ] for i = u).
Proof. The normally hyperbolic manifold and the foliation of its stable and
unstable manifolds behave smoothly under perturbation and equations (78–80) are a
simple consequence of this.
It remains to prove (81). Let i = s. We have
µ
¶
d ∂q s
∂ d s
(0, ϕ, t0 , 0, t) =
q (0, ϕ, t0 , 0, t)
dt ∂e
∂e dt
¡
¢2 ´
∂ ³
f (µ, q s (0, ϕ, t0 , e, t)) + eg (q s (0, ϕ, t0 , e, t), t) + O( eµ−1/3 ) |e=0
∂e
∂q s
(0, ϕ, t0 , 0, t) + g (q s (0, ϕ, t0 , 0, t)) .
= Df (µ, q s (0, ϕ, t0 , 0, t))
∂e
=
The points ps (0, ϕ, t0 , e) lie on stable fibres of the periodic orbit perturbed from Lµ2 .
The points ps (0, ϕ, t0 , e) and ps (0, ϕ, t0 , 0) are therefore O(e) close. Also the periodic
orbit perturbed from Lµ2 lies O(e) close to Lµ2 . Since orbits q s (0, ϕ, t0 , e, t) start from
ps (0, ϕ, t0 , e) this gives us
|q s (0, ϕ, t0 , e, t) − q s (0, ϕ, t0 , 0, t)| = O(e)
(82)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
30
for t ∈ [t0 , +∞). This means that
¯ ¯
¯
¯ s
¯ ¯
¯ ∂q
q s (0, ϕ, t0 , e, t) − q s (0, ϕ, t0 , 0, t) ¯¯
¯
¯
¯
lim
¯ ∂e (0, ϕ, t0 , 0, t)¯ = ¯e→0
¯
e
is bounded.
For i = u the argument is analogous.
Remark 36 Let us note that the bound O(r) in (78) is independent of µ.
Proof. This follows from formulas for the parameterization of the intersection of
the manifolds with {y = 0} from Theorem 4. Each curve from Theorem 4 gives an
intersection of an unstable manifold of a Lyapounov orbit. A point q i (r, ϕ, t0 , 0, t0 ) is a
point of intersection of the unstable manifold of a³Lyapounov orbit l(r) with
´ {y = 0},
√
√
and in x, ẋ coordinates is represented by a point x(σ, ∆C), ẋ(σ, ∆C) on a curve
from Theorem 4, for some σ ∈ S 1 and ∆C > 0. The point q i (0, ϕ, t0 , 0, t0 ) is the point of
intersection of the homoclinic orbit to Lµ2 k with {y = 0} and in x, ẋ coordinates is given
by (x(σ, 0), ẋ(σ, 0)) (where the choice of σ plays no role since for ∆C = 0 equations
from Theorem 4 give a single point).
From Theorem 4
√
√
√
√
(83)
x(σ, ∆C) − x(σ, 0) = O( ∆C) and ẋ(σ, ∆C) − ẋ(σ, 0) = O( ∆C).
By (16) and Lemma 17
√
q
H(µ, µ1/3 l(r) + (µ − 1, 0, 0, µ − 1)) − H(µ, Lµ2 )
q
= µ2/3 H̄(µ, l(r)) − µ2/3 H̄(µ, l(0))
∆C =
= µ1/3 O(r).
(84)
√
√
The points x(σ, ∆C), x(σ, 0), ẋ(σ, ∆C), ẋ(σ, 0) are considered
in original coordinates
´
³
√
√
of the system on the section {y = 0}. For a point (x, ẋ) = x(σ, ∆C), ẋ(σ, ∆C) on
the section {y = 0} by (3) we have px = ẋ − y = ẋ and
p
p
py = 2 (H(Lµ2 k ) + ∆C + Ω(x, 0)) − p2x = 2H(Lµ2 k ) + µ1/3 O(r).
Recall that the points q i (r, ϕ, t0 , 0, t0 ), q i (0, ϕ, t0 , 0, t0 ) are given in Hill’s coordinates
(15). By (15), (83), (84)
¯ i
¯
¯q (r, ϕ, t0 , 0, t0 ) − q i (0, ϕ, t0 , 0, t0 )¯
¯
¯
√
¯
¯
= µ−1/3 ¯(x, 0, px , py ) (σ, ∆C) − (x, 0, px , py ) (σ, 0)¯ = O(r).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
31
Remark 37 We believe that with techniques similar to the ones used for the proof of
Theorem 4 in [20], but involving additionally terms coming from the perturbation from
the PRC3BP to the PRE3BP, it should be possible to prove that the terms o(e) and
O(r) from (79), (80) can be chosen independently of µk . Such statement requires a
detailed proof which is rather technical. We skip this intentionally since in later parts
of our argument it will turn out that even if this had been done by us, we still cannot
obtain uniform bounds for the radius of set on which we have structural stability for
the PRE3BP. This is due to the fact that we have not obtained uniform bounds for the
Melnikov integral (85) (see Remark 40). Such bounds seem even harder to obtain than
proving that the terms o(e) and O(r) from (79), (80) are independent of µk .
Remark 36 will allow us though to extract some information as to the radius for
which we shall have structural stability for the PRE3BP. It will turn out that this radius
−1/3
needs to converge to zero with µk going to zero at least fast enough so that µk R(µk )
is bounded. From our proof though it is not transparent how small exactly it shall need
to be chosen.
We now have the following lemma regarding the energy of the points ps (r, ϕ, t0 , e)
and pu (r, ϕ, t0 , e).
Lemma 38 Assume that µ is one of the parameters µk for which a homoclinic orbit
¡
¢
q̄µ0 k (t) = µ−1/3 qµ0 k (t) − (µk − 1, 0, 0, µk − 1) to L̄µ2 = µ−1/3 (Lµ2 − (µk − 1, 0, 0, µk − 1))
exists (See Theorem 3). For any two points p1 , p2 ∈ Λe with coordinates (r1 , ϕ1 , t0 ) and
(r2 , ϕ2 , t0 ) respectively, we have
H̄(µk , ps (r1 , ϕ1 , t0 , e)) − H̄(µk , pu (r2 , ϕ2 , t0 , e))
= H̄(µk , l(r1 )) − H̄(µk , l(r2 )) + eMµk (t0 )
+ O(e max {|r1 |, |r2 |}) + o(e),
where
Z
Mµk (t0 ) =
+∞
−∞
{H̄, Ḡ}(µk , q̄µ0 k (t − t0 ), t)dt.
(85)
Proof. Let · denote the scalar product and let ∆s and ∆u denote the following
functions
∂q s
(0, ϕ1 , t0 , 0, t)
∂e
∂q u
0
∆u (t, t0 ) := ∇H̄(µk , q̄µk (t − t0 )) ·
(0, ϕ2 , t0 , 0, t).
∂e
∆s (t, t0 ) := ∇H̄(µk , q̄µ0 k (t − t0 )) ·
Using the facts that H̄ = H̄r = H̄(µk , l(r)) is constant along the solutions q s (r, ϕ, t0 , 0, t)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
32
of the PRC3BP, from (79) and (77) we can compute
H̄(µk , q s (r1 , ϕ1 , t0 , e, t0 ))
=H̄(µk , q s (r1 , ϕ1 , t0 , 0, t0 ))
∂q s
(r1 , ϕ1 , t0 , 0, t0 ) + o(e)
+ e∇H̄(µk , q (r1 , ϕ1 , t0 , 0, t0 )) ·
∂e
µ s
¶
∂q
0
=H̄r1 + e∇H̄(µk , q̄µk (0) + O(r1 )) ·
(0, ϕ1 , t0 , 0, t0 ) + O(r1 ) + o(e)
∂e
∂q s
=H̄r1 + e∇H̄(µk , q̄µ0 k (0)) ·
(0, ϕ1 , t0 , 0, t0 ) + O(er1 ) + o(e)
∂e
=H̄r1 + e∆s (t0 , t0 ) + O(er1 ) + o(e),
(86)
s
and similarly one can show that
H̄(µk , q u (r2 , ϕ2 , t0 , e, t0 )) = H̄r2 + e∆u (t0 , t0 ) + O(er2 ) + o(e).
(87)
Let us stress that, by Lemma 35 we know that O(er1 ), O(er2 ) and o(e) are uniform with
respect to t0 .
Let us investigate the evolution of ∆s (t, t0 ) and ∆u (t, t0 ) in time. Let us concentrate
on the term ∆s (t, t0 ). Using (81), (77) and ∇H̄ = −Jf (see (30)) we can compute
µ
¶
d
d 0
∂q s
0
− (∆s (t, t0 )) = JDf (µk , q̄µk (t − t0 )) q̄µk (t − t0 ) ·
(0, ϕ1 , t0 , 0, t)
dt
dt
∂e
¢ d ∂q s
¡
+ Jf (µk , q̄µ0 k (t − t0 )) ·
(0, ϕ1 , t0 , 0, t)
(88)
dt ∂e
¢ ∂q s
¡
= JDf (µk , q̄µ0 k (t − t0 ))f (µk , q̄µ0 k (t − t0 )) ·
(0, ϕ1 , t0 , 0, t)
∂e
µ
¶
¢ ∂q s
¢
¡
¡
0
0
+ Jf (µk , q̄µk (t − t0 )) · Df µk , q̄µk (t − t0 )
(0, ϕ1 , t0 , 0, t)
∂e
¡
¢
+ Jf (µk , q̄µ0 k (t − t0 )) · g(µk , q̄µ0 k (t − t0 ), t)
¡
¢
= Jf (µk , q̄µ0 k (t − t0 )) · g(µk , q̄µ0 k (t − t0 ), t)
= −{H̄, Ḡ}(µk , q̄µ0 k (t − t0 ), t),
where the third equality comes from the fact that for any p, q ∈ R4
¡
¢
¡
¡
¢ ¢
JDf (µk , q̄µ0 k (t − t0 ))p · q + (Jp) · Df µk , q̄µ0 k (t − t0 ) q = 0,
s
(89)
with p = f (µk , q̄µ0 k (t − t0 )) and q = ∂q
(0, ϕ1 , t0 , 0, t). Equation (89) follows from the fact
∂e
that ω(p, q) = Jp · q is the standard symplectic form which is invariant under the flow
φ(t, x) of (16) i.e.
¶
µ
∂
∂
φ(t, x)p, φ(t, x)q = ω(p, q),
(90)
ω
∂x
∂x
hence by differentiating (90) with respect to t and setting t = 0 we obtain (89).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
33
We can now compute ∆s (t0 , t0 ) using (88)
Z +∞
∆s (+∞, t0 ) − ∆s (t0 , t0 ) =
{H̄, Ḡ}(µk , q̄µ0 k (t − t0 ), t)dt.
t0
Since limt→+∞ q̄µ0 k (t − t0 ) = L̄µ2 k at geometric rate and f (µk , L̄µ2 k ) = 0, from the fact that
∂q s
(0, ϕ1 , t0 , 0, t) is bounded on [t0 , +∞) we have
∂e
∆s (+∞, t0 ) = lim Jf (µk , q̄µ0 k (t − t0 )) ·
t→+∞
and therefore
Z
+∞
−∆s (t0 , t0 ) =
t0
∂q s
(0, ϕ1 , t0 , 0, t) = 0
∂e
{H̄, Ḡ}(µk , q̄µ0 k (t − t0 ), t)dt,
(91)
and ∆ is uniformly with respect to t0 absolutely convergent.
Analogous computations give
Z t0
∆u (t0 , t0 ) =
{H̄, Ḡ}(q̄µ0 k (t − t0 ), t)dt.
(92)
−∞
From (86), (87), (91) and (92) we obtain our claim.
Theorem 39 Consider the PRE3BP with a sufficiently small parameter µ = µk for
which a homoclinic orbit to Lµ2 k exists (See Theorem 3 ). Assume that
Z +∞
Mµk (t0 ) =
{H̄, Ḡ}(µk , q̄µ0 k (t − t0 ), t)dt
(93)
−∞
−1/3
has simple zeros. Let R(µk ) ∈ R, be such that 0 < R(µk ) and µk R(µk ) is
sufficiently close to zero. Then for any R ∈ (0, R(µk )) there exists an e0 (R) such
−1/3
1/3
that for all eµk
∈ [0, min(e0 (R), κµk )] (where κ is the constant from (25 )) and all
r ∈ C ∩ [R, R(µk )] for which l(r) is perturbed to an invariant torus le (t)
le (r) = {(lte0 (r), t0 )|t0 ∈ S 1 }
the manifolds Wlse (r) and Wlue (r) (considered in the extended phase space) intersect
transversally.
Proof. We consider the PRE3BP (29) in Hill’s coordinates (15). In [20] it has
been shown (see also Theorem 4 in Section 2.1) that for the unperturbed PRC3BP
s
u
Wl(r)
intersects transversally with Wl(r)
at {ȳ = 0}. Let Σt0 = R4 × {t0 } be the time t0
section in the extended phase space. Let v 0 (t0 ) denote some point for which
s
u
v 0 (t0 ) ∈ Wl(r)
∩ Wl(r)
∩ {ȳ = 0} ∩ Σt0 .
(94)
There can be more than just one such point (see Figures 3, 4), namely, if we consider
u
s
πx,px (Wl(r)
∩ Σt0 ∩ {ȳ = 0}) and πx̄,p̄x (Wl(r)
∩ Σt0 ∩ {y = 0}) then the two sets
are homeomorphic to two circles, which intersect transversally at least one point
Transition Tori in the Planar Restricted Elliptic Three Body Problem
34
u
s
, Wlse (r) and Wlue (r) for r > R, intersected with
, Wl(r)
Figure 4. The manifolds Wl(r)
Σt0 and {y = 0}, and projected onto the x, px coordinates.
(x̄ = x̄0 , p̄x = 0) (see Theorem 4). Fixing any one of such points of intersection will be
sufficient for our proof. From the construction we have πx,y,px ,t (v 0 (t0 )) = (x̄0 , 0, 0, t0 ).
The extended phase space of the PRC3BP is five dimensional. We can choose the
energy H̄ to be the remaining fifth coordinate in the neighborhood of v 0 (t0 ). In these
coordinates
¡
¢
v 0 (t0 ) = x̄ = x̄0 , ȳ = 0, p̄x = 0, H̄ = H̄(µk , l(r)), t = t0 .
Now we perturb from the PRC3BP to the PRE3BP. Consider sufficiently small µk
−2/3
and perturbation e satisfying eµk
< κ so that by Theorem 31 we have a Cantor set C
e
of KAM tori l (r)L̇et us consider some small R(µk ) ∈ R (the size of R(µk ) needs to be
chosen small compared with Mµk (t). This is discussed later on in our argument). Let
0 < R < R(µk ) and r ∈ C ∩ [R, R(µk )]. Consider now the following sets
¡
¢
¡
¢
πx̄,p̄x Wlue (r) ∩ Σt0 ∩ {ȳ = 0}
and πx̄,p̄x Wlse (r) ∩ Σt0 ∩ {ȳ = 0} .
For sufficiently small e these sets remain homeomorphic to circles. For e = 0 the
1/3
curves intersect transversally at an angle O(µk ) (see Remark 7). This means that
by choosing eµ−1/3 > 0 sufficiently small the curves shall intersect at some point
(x̄ = x̄0 (t0 , e), p̄x = p̄0x (t0 , e)) which is close to (x0 , 0). The choice of eµ−1/3 also
depends on R, since in order to ensure that the curves intersect we assume that
their radius is greater than R. Hence we have e0 (R) in the formulation of our
theorem. For the PRE3BP the energy H̄ is no longer preserved. This means that
the intersection of the circles on the x̄, p̄x plane does not imply an intersection in the
extended phase space. Namely we have two points v s (t0 , e) ∈ Wlse (r) ∩ Σt0 ∩ {ȳ = 0} and
v u (t0 , e) ∈ Wlue (r) ∩ Σt0 ∩ {ȳ = 0} which may differ on the energy coordinate (see Figure
5)
¢
¡
(95)
v s (t0 , e) = x̄0 (t0 , e), 0, p̄0x (t0 , e), hs (t0 , e), t0
¢
¡
v u (t0 , e) = x̄0 (t0 , e), 0, p̄0x (t0 , e), hu (t0 , e), t0 .
We will show that assumptions of our theorem imply that for some t0 the points v s (t0 , e)
and v u (t0 , e) coincide. This will imply intersection between Wlse (r) and Wlue (r) . Later we
will also show that such intersection is transversal.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
35
The points v s (t0 , e) and v u (t0 , e) are both contained in {ȳ = 0}. Moreover
v s (t0 , e) ∈ Wlse (r) ∩ Σt0 = Wlste (r) and v u (t0 , e) ∈ Wlue (r) ∩ Σt0 = Wlute (r) . This means
0
0
that there exist rs , ϕs and ru , ϕu (these depend on t0 and e but we omit this in our
notations for simplicity) such that
v s (t0 , e) = (ps (rs , ϕs , t0 , e), t0 )
u
u
u
(96)
u
v (t0 , e) = (p (r , ϕ , t0 , e), t0 ).
In the proof of Theorem 31 lte0 (r) is constructed from continuation along trajectories of
a KAM torus l0e (r). Therefore from (72) in KAM Theorem 29 we have that
rs = r + O(eµ−1/3 ),
u
−1/3
r = r + O(eµ
(97)
),
and the bound O(eµ−1/3 ) is uniform for all r ∈ C. Applying Lemma 38 we have that
hu (t0 , e) − hs (t0 , e) = H̄(µk , ps (rs , ϕs , t0 , e)) − H̄(µk , pu (ru , ϕu , t0 , e))
(98)
= H̄(µk , l(rs )) − H̄(µk , l(ru )) + eMµk (t0 ) + O(eR(µk )) + o(e).
The Hamiltonian in coordinates (q, p) = (q1 , q2 , p1 , p2 ) = (x̄, ȳ, p̄x , p̄y ) − L̄µ2 k centered in
L̄µ2 k is
¢
¡
H (p, q) := H̄ µk , (q, p) + L̄µ2 k .
By Lemma 17 we hence know that
1
H̄(µk , l(r)) = H̄(µk , L̄µ2 k ) + D2 H(0) (Φ(0, 1, 0, i)) r2 + o(r2 )
2
hence from (97) and (98) we have
hu (t0 , e) − hs (t0 , e) = O(reµ−1/3 ) + eMµk (t0 ) + O(eR(µk )) + o(e)
−1/3
= eMµk (t0 ) + O(eµ
(99)
R(µk )) + o(e).
−1/3
Setting first R(µk ) sufficiently small (so that µk R(µk ) is small in comparison to
Mµk (t)) and then reducing e sufficiently close to zero implies that since Mµk (t0 ) has
simple zeros, for some parameters t0 (close to these zeros) we will have hu (t0 , e) −
hs (t0 , e) = 0, which implies that v s (t0 , e) = v u (t0 , e) and in turn ensures that
Wlse (r) ∩ Wlue (r) 6= ∅.
Now we will show that this intersection is transversal. First note that from the
analyticity of the functions v s (t0 , e) and v u (t0 , e) using the same argument as in the
proof of Lemma 38 we also have that
∂
∂ u
−1/3
(h (t0 , e) − hs (t0 , e)) = e Mµk (t) + O(eµk R(µk )) + o(e).
∂t
∂t
(100)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
36
s
u
We know that prior to our perturbation Wl(r)
and Wl(r)
intersect transversally at v 0 (t0 )
(94). This intersection is not transversal in the full extended phase space, it is only
transversal in the constant energy manifold
M = {(x̄, ȳ, p̄x , H̄, t)|H̄ = H̄(µk , l(r))} ⊂ R3 × {H̄(µk , l(r))} × S 1 .
To be more precise, we know that v 0 (t0 ) ∈ Wls0
t0 (r)
and that [20]
Tv0 (t0 ) (Wlst0
0
(r) )
∩ Wlu0 (r) , that Wls0
t0
t0 (r)
, Wlu0 (r) ⊂ Σt0
t0
+ Tv0 (t0 ) (Wlut0 (r) ) = R3 × {0} × {0}.
0
These properties are preserved under small perturbation e > 0, hence for v s (t0 , e) =
v u (t0 , e) =: v(t0 , e) we have
Tv(t0 ,e) (Wlste
0
(r) )
+ Tv(t0 ,e) (Wlute (r) ) = R3 × {0} × {0}.
0
We need to show that we also have transversality on the t0 and energy coordinate. For
a fixed e the curves v s (t, e) and v u (t, e) belong to Wlse (r) and Wlue (r) respectively. At the
time t = t0 for which v s (t0 , e) = v u (t0 , e) = v(t0 , e) we have Mµ0 k (t0 ) 6= 0. This means
that using (100), for sufficiently small e,
∂
∂ u
(πH (v s (t, e)) − πH (v u (t, e))) |t=t0 =
(h (t, e) − hs (t, e)) |t=t0
∂t
∂t
−1/3
= eMµ0 k (t0 ) + O(eµk R(µk )) + o(e)
6= 0.
∂
∂
(πt0 v i (t, e)) = ∂t
t = 1. This since dtd v s (t, e)|t=t0 ∈
We also have for i ∈ {u, s}, ∂t
Tv(t0 ,e) (Wlse (r) ) and dtd v u (t, e)|t=t0 ∈ Tv(t0 ,e) (Wlue (r) ) implies transversality, which finishes
our proof.
The order of choice of parameters in the above argument is important, so let us
quickly run through how it should be conducted. We first choose sufficiently small µk so
−1/3
that we can apply Theorem 31. Then by choosing small R(µk ) we ensure that µk R(µk )
−1/3
is sufficiently small compared with Mµk and Mµ0 k . We then choose e so that eµk
is
u
sufficiently small so that we have transversal intersections of πx,px (Wle (r) ∩ Σt0 ∩ {ȳ = 0})
and πx,px (Wlse (r) ∩ Σt0 ∩ {ȳ = 0}). The parameter e needs also to be sufficiently small so
that Mµk (t0 ) and Mµ0 k (t0 ) dominate in (99) and (100) respectively.
Remark 40 In the proof of Theorem 39 we see that we need to choose the radius R(µk )
−1/3
to be sufficiently small so that µk R(µk ) is small in comparison to the Melnikov integral
Mµk and its derivative. Since we do not have uniform bounds on the size of the Melnikov
integral with respect to µk , from our argument we cannot say how small R(µk ) needs to
be. From our numerical investigation which will follow in Table 2 we can see that for
t0 for which we have a simple zero of the Melnikov integral, the bound on the derivative
1/3
is independent of µk . This means that we need to choose the radius R(µk )µk to be
sufficiently small (hence R(µk ) converges to zero with µk going to zero).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
37
Figure 5. The manifolds Wlse (r) and Wlue (r) intersected with Σt1 , Σt2 and {y = 0}, in
the x, px , H coordinates for t1 < t0 < t2 .
Corollary 41 If µk is sufficiently small and the Melnikov integral has a simple zero
−1/3
then for sufficiently small µk R(µk ) there exists a ζ > 0, such that for any two radii
r1 and r2 from C ∩ [R, R(µk )]
−1/3
|r1 − r2 | < ζeµk
,
the manifolds Wlse (ri ) and Wlue (rj ) intersect transversally for i, j ∈ {1, 2}.
Proof. The proof of this fact is a mirror argument to the proof of Theorem 39.
Below we restrict our attention to pointing out the difference we have connected with
the derivation of (99) in the setting where we have two radii.
Let hu1 (t0 , e) and hs2 (t0 , e) stand for energies of points of potential intersection
(constructed analogously to hu (t0 , e) and hs (t0 , e) in (95)). Let r1u and r2s stand for
radii constructed analogously to ru and rs (see (96)), but coming from the unstable
and stable manifold of le (r1 ) and le (r2 ) respectively. Using Lemma 17 and the fact that
−1/3
−1/3
r1u − r1 = O(eµk ) and r2s − r2 = O(eµk ) we have
|H̄(µk , l(r1u )) − H̄(µk , l(r2s ))| ≤ O(|(r1u )2 − (r2s )2 |)
≤ O(R(µk )|r1u − r2s |)
−1/3
≤ O(R(µk )|r1 − r2 |) + O(eµk
R(µk )).
Using an identical argument to the derivation of (99) this gives us
−1/3
hu1 (t0 , e) − hs2 (t0 , e) = eMµk (t0 ) + O(R(µk )|r1 − r2 |) + O(eµk
R(µk )) + o(e).
Using this estimate and following the proof of Theorem 39 we obtain our claim.
Remark 42 Mirror arguments to the proof of Theorem 39 and Corollary 41 give
transversal intersections of invariant manifolds for invariant tori of the PRE3BP with
µ = µk for k = 2, 3, . . .. Here we state this as a separate remark since Theorem 39 and
Corollary 41 are fully rigorous and do not rely on any numerical computations. For the
argument with an arbitrary µk we would need to use the fact that in the PRC3BP we
have a twist property on the family of Lyapounov orbits and apply Remark 32. The twist
for arbitrary µk has only been demonstrated numerically (see Table 1).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
38
7. Computation of the Melnikov integral.
In this section we will demonstrate that for t0 = 0 and for all parameters µk from
dM
Theorem 3 the Melnikov integral Mµk (t0 ) (93) is zero and also that dtµ0 k (0) 6= 0. The
fact that the Melnikov integral is zero will follow directly from the S-symmetry (8) of the
dM
homoclinic orbit qµ0 k (t). The fact that dtµ0 k (0) 6= 0 will be demonstrated numerically.
We will compute the integral for the first few parameters µk and then demonstrate
that for sufficiently small parameters µk the integral converges to an integral along an
unstable manifold of the Hill’s problem.
7.1. The Melnikov integral and its derivative at t0 = 0
We start with a lemma which ensures the convergence of the Melnikov integral (93).
Lemma 43 The Melnikov integral (93) and its derivative is absolutely convergent
uniformly with respect to t0 . The Melnikov function can be expressed as
¸
Z +∞ ·
¢ ∂ Ḡ ¡
¢
∂ Ḡ ¡
µk
0
Mµk (t0 ) =
(101)
µk , q̄µk (t) , t + t0 −
µk , L̄2 , t + t0 dt,
∂t
∂t
−∞
and also
dMµk
(t0 ) =
dt
Z
+∞
−∞
·
¸
¢ ∂ 2 Ḡ ¡
¢
∂ 2 Ḡ ¡
µk
0
µk , q̄µk (t) , t + t0 − 2 µk , L̄2 , t + t0 dt.
∂t2
∂t
(102)
Proof. The orbit q̄µ0 k (t) is the homoclinic orbit to the Libration point L̄µ2 k . Let us
note that the velocity x̄0 and ȳ 0 of q̄µ0 k (t) exponentially tends to zero as t tends to plus
infinity and minus infinity. Moreover the partial derivatives of Ḡ on q̄µ0 k (t) are uniformly
bounded. This means that the integral over
Z +∞
Z +∞
∂ Ḡ
∂ Ḡ
0
|x̄0
+ ȳ 0
|(µk , q̄µ0 k (t), t + t0 )dt,
|{H̄, Ḡ}|(q̄µk (t), t + t0 )dt =
∂ x̄
∂ ȳ
−∞
−∞
is convergent uniformly with respect to t0 .
The orbit q̄µ0 k (t) is the solution of the PRC3BP, hence differentiating gives
¢
¢ ∂ Ḡ ¡
¢ ©
ª¡
dḠ ¡
µk , q̄µ0 k (t) , t + t0 =
µk , q̄µ0 k (t) , t + t0 + Ḡ, H̄ µk , q̄µ0 k (t) , t + t0 . (103)
dt
∂t
From (103) we have
Z +∞
©
ª
H̄, Ḡ (µk , q̄µ0 k (t), t + t0 )dt
Mµk (t0 ) =
−∞
¶
Z T µ
¢ dḠ ¡
¢
∂ Ḡ ¡
0
0
= lim
µk , q̄µk (t) , t + t0 −
µk , q̄µk (t) , t + t0 dt
T →∞ −T
∂t
dt
¢
¢
¡
£ ¡
= lim Ḡ µk , q̄µ0 k (−T ) , −T + t0 − Ḡ µk , q̄µ0 k (T ) , T + t0
T →∞
¸
Z T
¢
∂ Ḡ ¡
0
+
µk , q̄µk (t) , t + t0 dt .
−T ∂t
(104)
Transition Tori in the Planar Restricted Elliptic Three Body Problem
39
To complete the proof of (101) observe that that from limT →±∞ q̄µ0 k (T ) = L̄µ2 k it follows
that
µZ T
¢
¡
¢
∂ Ḡ ¡
lim
µk , L̄µ2 k , t + t0 dt − Ḡ µk , q̄µ0 k (T ) , T + t0
(105)
T →∞
−T ∂t
¶
¡
¢
0
+ Ḡ µk , q̄µk (−T ) , −T + t0
=0
uniformly with respect to t0 .
From (104) and (105) we obtain (101).
To prove (102) it is enough to observe that the formal integration of (101) is correct,
because the integral on the right hand side of (102) is uniformly convergent with respect
to t0 for the same reasons as the integral in formula (101).
It turns out that the computation of the Melnikov integral is not the major obstacle.
The fact that we have a zero for t0 = 0 follows directly from the S-symmetry (8) of the
homoclinic orbit qµ0 k (t) and G. This fact is shown in the below lemma . Later though
dM
we will need to show that this zero is nontrivial by computing dtµk (0), which turns out
to be a much harder task. The lemma also provides the formula for the needed integral.
Lemma 44 The Melnikov integral (93) at t0 = 0 is equal to zero and
Z 0
¢
¢
¡ ¡
dMµk
−2/3
(0) = −2µk
G µk , qµ0 k (t) , t − G (µk , Lµ2 k , t) dt.
dt
−∞
(106)
Proof. First let us observe that from (27)
¢
¢
¡
−2/3 ¡
Ḡ µk , q̄µ0 k (t) , t = µk G µk , qµ0 k (t) , t .
The orbit qµ0 k (t) and fixed point Lµ2 k are S-symmetric with respect to the symmetry (8).
(µ, S (·) , −t) = − ∂G
(µ, ·, t)
From (7), (23), by direct computation one can check that ∂G
∂t
∂t
2
2
∂ G
∂ G
and that ∂t2 (µ, S (·) , −t) = ∂t2 (µ, ·, t) hence we have
¶
Z 0 µ
¢ ∂ Ḡ ¡
¢
∂ Ḡ ¡
µk
0
µk , q̄µk (t) , t −
µk , L̄2 , t dt =
∂t
∂t
−∞
¶
Z +∞ µ
∂G
∂G
−2/3
µk
0
= µk
(µk , qµk (−t), −t) −
(µk , L2 , −t) dt
∂t
∂t
0
¶
Z +∞ µ
¡ 0
¢
¢ ∂G
∂G ¡
−2/3
µk
µk , S qµk (t) , −t −
(µk , S (L2 ) , −t) dt
= µk
∂t
∂t
0
¶
Z +∞ µ
¢ ∂G
∂G ¡
−2/3
µk
0
= −µk
µk , qµk (t) , t −
(µk , L2 , t) dt,
∂t
∂t
0
2
which gives Mµk (0) = 0. From an analogous computation using ∂∂tG2 (S (·) , −t) =
∂2G
(·, t) follows
∂t2
¶
Z 0 µ 2
¢ ∂ 2G
∂ G¡
dMµk
−2/3
µk
0
(0) = 2µk
µk , qµk (t) , t − 2 (µk , L2 , t) dt.
dt
∂t2
∂t
−∞
Form (23) we have
∂2G
∂t2
= −G, which gives (106).
Transition Tori in the Planar Restricted Elliptic Three Body Problem
40
dM
Remark 45 The verification of the fact that dtµk (0) is nonzero is not straightforward.
In this paper we will restrict ourselves to numerical verification of this fact. We would
like to highlight that for a given parameter µk it is possible to obtain a rigorous-computerdM
assisted estimate on dtµk (0). To do so one first needs to obtain a rigorous bound [µk , µk ]
which contains the parameter µk for which we have a homoclinic orbit. Then one needs
to obtain rigorous enclosures on the trajectories qµ0 (t) for all µ ∈ [µk , µk ]. Using these,
dM
a bound on dtµk (0) can be computed from (106).
We have successfully conducted such computations for the parameter µ2 . We have
used the fact that if one extends the system by including µ as an additional variable,
then for any interval I the set {Lµ2 }µ∈I is a normally hyperbolic invariant manifold.
We have applied a topological method, given in [6], [7], for detection of normally
hyperbolic manifolds, combined with a parameterization method [5]. Based on these
we proved the following. We have shown that the parameter µ2 for which we have
the homoclinic orbit is contained in [0.0042538631, 0.0042538639]. We then proved that
dMµ2
(0) ∈ [1.301020122, 1.865308899]. A detailed proof of this fact, along with results
dt
for other parameters, will be the subject of a forthcoming publication.
In Table 2 we enclose the (nonrigorous) numerical results for the computation of
(106) obtained for k up to 13.
k
2
3
4
5
6
7
8
9
10
11
12
13
dMµk
(0)
dt
1.57396
-0.396727
0.395119
-0.396931
0.395784
-0.395511
0.393389
-0.393253
0.390924
-0.391194
0.388961
-0.389459
Table 2. Numerical results for the derivation of
dMµk
(0)
dt
for various mass parameters.
To obtain the numerical results from Table 2 we have used a parameterization
method [5] to obtain an expansion of the manifold around Lµ2 k as a polynomial of degree
20. Then we integrated the system numerically using a Taylor method of order 20.
Numerical evidence points to Mµk having a nontrivial zero for t = 0.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
41
8. Transition Chains and Main Result
In this section we will show that there exists a sequence of Lyapounov orbits l(ri ) for
i = 1, . . . , N which survive for a sufficiently small perturbation e, and such that their
stable and unstable manifolds intersect transversally
Wlse (ri ) t Wlue (ri+1 )
and Wlue (ri ) t Wlse (ri+1 ) .
(107)
Such sequences are referred to as transition chains. Existence of such chains ensures
that we have an orbit which shadows the homo- or heteroclinic connections between
the surviving tori generating rich symbolic dynamics (see [10],[11],[12]). Apart from
this, from our argument it will also follow that for any surviving orbit l(r) we also have
transversal intersection of its stable and unstable manifold
Wlue (r) t Wlse (r) .
(108)
This ensures that the chaotic dynamics of the PRC3BP which is implied by the existence
of a transversal homoclinic orbit to l(r) (see Theorem 4 and Remark 6) survives.
We are now ready to rigorously reformulate our main theorem (Theorem 2).
Theorem 46 For any µ from the sequence of masses {µk }∞
k=2 from Theorem 3, if the
twist property is satisfied, then there exists a radius RHill , which is independent of µ, such
that for eccentricities e of the elliptic problem, with eµ−2/3 < κ (see 25 for interpretation
of κ) and sufficiently small eµ−1/3 , there exists a Cantor set C ⊂ [0, RHill ] such that for
all r ∈ C the Lyapounov orbits l(r) are perturbed to invariant tori le (r) in the extended
phase space. Moreover, if the derivative of the Melnikov integral (106) is nonzero, then
there exists a radius R(µ) of order at most O(µ1/3 ) and a transition chain le (ri ) for a
sequence of radii 0 < r1 < r2 < . . . < rN < R(µ) such that the difference rN − r1 is of
¡
¢1/2
order eµ−1/3
.
The transversal intersections of the transition chain (107), (108) lead to a homoand heteroclinic tangle of the stable/unstable manifolds of le (ri ), which in turn leads to
symbolic dynamics involving diffusion in energy.
Remark 47 Let us note now that we can verify assumptions of the above theorem
based on some numerical results. The twist property for the family of Lyapounov orbits
has been rigorously proved for sufficiently small µk in Theorem 31, but the fact that
we have twist for µk with k = 1, 2, . . . has only been demonstrated numerically (see
Section 4, Table 1 and Remark 32). Secondly, assumptions of Theorem 46 require that
the derivative of the Melnikov function (106) at zero is nonzero. This has only been
demonstrated numerically in Section 7.
We believe that the above can be verified using rigorous-computer-assisted methods.
This is currently a subject of ongoing work (see Remark 45).
Remark 48 All radiuses considered in Theorem 46 are given in the Hill’s coordinates
(15). This means that in the original coordinates of our system (given by the
Hamiltonian (22)) the radiuses are reduced by a factor of µ1/3 .
Transition Tori in the Planar Restricted Elliptic Three Body Problem
42
Proof of Theorem 46. Let us fix a µ = µk . For sufficiently small µ by Theorem
31, and if the twist condition holds for all µk by Remark 32, we have the radius
RHill and a Cantor C ⊂ [0, RHill ] of radii for which the Lyapounov orbits survive the
perturbation. From Theorem 39, Remark 42 (see also Corollary 41) it follows that by
choosing R(µ) < RHill , for which R(µ)µ−1/3 is sufficiently small, we know that there
exists a ζ > 0 such that for radii r1 , r2 ∈ C such that |r1 − r2 | < ζeµ−1/3 , for e with
sufficiently small eµ−1/3 , we have
Wlue (ri ) t Wlse (ri )
for i ∈ {1, 2}.
We now need to show that we can find a sequence r1 < r2 < . . . < rN such that ri ∈ C
¡
¢1/2
and the difference rN − r1 is of order eµ−1/3
, for which the gaps between ri and ri+1
−1/3
are smaller than ζeµ
. The existence of such a sequence follows from Proposition 34.
The last claim of Theorem 46 follows from [10], [11].
9. Concluding remarks, future work
In this paper we have shown that the chaotic dynamics observed for the planar restricted
circular three body problem survives the perturbation into the planar restricted elliptic
three body problem, when its eccentricity is sufficiently small. We have also shown that
this dynamics is extended to include diffusion in energy. The diffusion proved in this
paper covers a small range of energies. This is due to the fact that in our argument we
use a Melnikov type method which does not allow us to jump between the ”large gaps”
between the KAM tori. An interesting problem which could be addressed is whether
these large gaps can be overcome (this potentially could be done using techniques similar
to [9] or [12]).
Our result holds only for a specific family {µk } of masses of the primaries. The
choice of these masses is such that they ensure the existence of the homoclinic orbit to
Lµ2 k , which is then used for the Melnikov argument. An interesting question is whether
one can observe similar dynamics in real life setting, say in the Jupiter-Sun system. In
such a case we will no longer have a homoclinic connection for the point Lµ2 . For the
(circular) Jupiter-Sun system though we know that we have a transversal homoclinic
connection for Lyapounov orbits (see [17] and [28]). Such orbits could be used for
a similar construction. Our argument also required that we have sufficiently small
eccentricities. It would be interesting to find out if the dynamics persists for the actual
eccentricity of the Jupiter-Sun system. For this problem it is quite likely that applying
the mechanism discussed in this paper would be very hard. Our argument relies on the
use of the KAM theorem, which works for sufficiently small perturbations. To apply
it for an explicit eccentricity seems a difficult task. Other methods could be exploited
though. Instead of proving the persistence of the tori and trying to detect intersections
of their invariant manifolds, one could focus on detection of symbolic dynamics for the
diffusing orbits in the spirit of [28]. This seems a far more realistic target for the near
future and is being currently considered as an extension of this work.
Transition Tori in the Planar Restricted Elliptic Three Body Problem
43
10. Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments
and suggestions which helped them improve the quality of the paper.
11. Appendix
11.1. Splitting of manifolds associated to Lyapounov orbits of the PRC3BP
Here we investigate the dependence on parameter µ of the splitting of the intersection
of curves from Theorem 4. Let µ = µk , which means that in (12) α = 0. The transversal
intersection of the curves is on {ẋ = 0}. The curve intersects {ẋ = 0} for σ = σ0 for
which
(K1 cos τ cos σ0 − K2 sin τ sin σ0 ) = O(µ4/3 ).
(109)
This is because from (12) only then can we have Mf (σ0 ) sufficiently close to zero so that
ẋ = 0. In particular, (12) and (109) gives
Mf (σ0 ) = O(µ2/3 ).
(110)
By implicit function theorem we know that
√
∂M
γ(µ, ∆C) :=
(σ0 )
(111)
∂σ √
µ−2/3 ∆C3M (N + 2M cos τ )−1 (−K1 cos τ sin σ0 − K2 sin τ cos σ0 )
=−
N + 2M cos(Mf (σ0 ))
√
∈ µ−2/3 ∆C [a1 , a2 ] ,
where a1 , a2 are constants, 0 ∈
/ [a1 , a2 ] (for sufficiently small µ and ∆C these constants
are arbitrarily close to one another. For their precise values one would need to substitute
the constants M, τ , K1 , K2 , N from [20] into (111)). By (109), (110), (111)
∂x
(σ0 )
∂σ
³
´
√
√
= ∆C (N + 2M cos τ )−1 N − sin (Mf (σ0 )) γ(µ, ∆C)
· (K1 cos τ cos σ0 − K2 sin τ sin σ0 )
√
+ ∆C (N + 2M cos τ )−1 (2M + N cos Mf (σ0 )) (−K1 cos τ sin σ0 − K2 sin τ cos σ0 )
√
+ µ1/3 M sin (Mf (σ0 )) γ(µ, ∆C)
¾
½
√
√
2M N
2
2/3
sin (Mf (σ0 )) γ(µ, ∆C) + M 2 sin (Mf (σ0 )) cos (Mf (σ0 )) γ(µ, ∆C)
+µ
−
3
√
∈ ∆C [b1 , b2 ] ,
Transition Tori in the Planar Restricted Elliptic Three Body Problem
44
where b1 , b2 are constants, 0 ∈
/ [b1 , b2 ] (the second term in the above equation plays a
dominant role). Also
"√
√
∂ ẋ
∆CN (K1 cos τ cos σ0 − K2 sin τ sin σ0 )
(σ0 ) = cos (Mf (σ0 )) γ(µ, ∆C)
∂σ
N + 2M cos τ
¾¸
½
M
MN
1/3
2/3
2
+µ M + µ
+ 2M cos Mf + α
3
3
√
−1
+ sin (Mf (σ0 )) [ ∆CN (N + 2M cos τ ) (−K1 cos τ sin σ0 − K2 sin τ cos σ0 )
√
− 2M 2 sin (Mf (σ0 )) γ(µ, ∆C)]
√
∈ µ−1/3 ∆C[c1 , c2 ],
where c1 , c2 are constants, 0 ∈
/ [c1 , c2 ] . This means that
∂x
∂ ẋ
(σ0 )/
(σ0 ) ∈ µ1/3 [d1 , d2 ],
∂σ
∂σ
(112)
for constant d1 , d2 , 0 ∈
/ [d1 , d2 ] . Stable/unstable manifolds are S symmetric (see (8) and
Theorem 4), hence for a curve coming from the intersection of the stable manifold with
∂x
∂ ẋ
{y = 0} we shall have same estimates for ∂σ
, and estimates with reversed sign for ∂σ
.
1/3
This by (112) gives a splitting of the two manifolds with angle of order µ .
11.2. Derivation of equations for the PRE3BP in rotating coordinates
Here we derive the Hamiltonian (22) of the PRE3BP in rotating coordinate system. Let
R(ϕ) be the rotation by the angle ϕ
"
#
cos ϕ − sin ϕ
R(ϕ) =
.
sin ϕ cos ϕ
The ellipse which is obtained when solving the 2-body problem is given by (see (21))
r(t) =
1 − e2
= 1 − e cos ψ(t) + O(e2 ),
1 + e cos ψ(t)
(113)
z(t) = r(t)R(ψ(t)) · [1, 0]T ,
ψ(t) = t + 2e sin t + O(e2 ).
The primary with the mass µ (the planet) has the following location zp (t) = (µ − 1)z(t),
while the primary mass 1 − µ (the Sun) is located at the point zs (t) = µz(t).
We shall now compute the distances between the comet and the primaries in
rotating coordinates. Let r1 (t) be the square of the distance between the comet and the
Sun, and r2 (t) between the comet and the planet. Let (x(t), y(t)) denote the ’rotating’
coordinates of the comet and q(t) be the position of the comet in the ’static’ coordinate
frame (x(t), y(t)) = R(−t)q(t). Using the fact the the length of the vector is not changed
Transition Tori in the Planar Restricted Elliptic Three Body Problem
45
by the rotation and (113) we have
°
°2
r1 (t)2 = °R(t) · (x, y)T − µr(t)R(ψ(t)) · [1, 0]T °
°
°2
= °(x, y)T − µr(t)R(ψ(t) − t) · [1, 0]T °
= x2 + y 2 − 2µr(t) (x cos (ψ(t) − t) + y sin (ψ(t) − t)) + µ2 r(t)2
= x2 + y 2 − 2µ (x cos (ψ(t) − t) + y sin (ψ(t) − t)) + µ2 +
2µe cos ψ(t) (x cos (ψ(t) − t) + y sin ψ(t) − t)) − 2µ2 e cos ψ(t) + O(µe2 ).
Using
ψ(t) = t + 2e sin t + O(e2 ),
cos (ψ(t)) = cos t − 2e sin2 t + O(e2 ),
cos (ψ(t) − t) = 1 + O(e2 ),
sin (ψ(t) − t) = 2e sin t + O(e2 ),
x cos (ψ(t) − t) + y sin (ψ(t) − t) = x + 2ey sin t + O(e2 ),
cos (ψ(t)) (x cos (ψ(t) − t) + y sin (ψ(t) − t)) = x cos t + O(e),
in the expression for r1 (t) we obtain
r1 (t)2 = (x − µ)2 + y 2 + 2eḡ(µ, x, y, t) + O(µe2 ),
(114)
where ḡ is given in (24). Expression for r2 (t) is obtained from (114) with the substitution
µ 7→ (µ − 1). Observe that
µ
µ³ ´ ¶¶
1
c 2
1
1
c
√
= p
=
1
−
+
O
.
(115)
r
2r2
r2
r 1 + rc2
r2 + c
We shall use notation r1 , r2 from (4), with r1 > δ and r2 > µ1/3 δ. Note that to apply
(115) for r = r2 we need to have eḡ(µ − 1, x, y, t) + O(µe2 ) < r22 , which means that we
need to take e sufficiently small so that eµ−2/3 < κ for sufficiently small κ. Equations
(114), (115) give
1
1
eḡ(µ, x, y, t)
=
−
+ O(µ2 e2 ),
r1 (t)
r1
r13
1
eḡ(µ − 1, x, y, t)
1
=
−
+ O(e2 µ−5/3 ).
r2 (t)
r2
r23
(116)
(117)
Substituting (116), (117) into (1) gives (22).
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